{\rtf1\ansi\ansicpg1252\uc1\htmautsp\deff2{\fonttbl{\f0\fcharset0 Times New Roman;}{\f2\fcharset0 Microsoft Sans Serif;}{\f3\fcharset0 Trebuchet MS;}}{\colortbl\red0\green0\blue0;\red255\green255\blue255;}\loch\hich\dbch\pard\tx1700\tx3400\tx5100\plain\ltrpar\itap0{\lang1033\fs40\f3\cf0 \cf0\ql{\f3\b {\ltrch Topology}\li0\ri0\sa160\sb0\fi0\ql\par} {\f3 {\ltrch In the previous chapters we dealt with collections of points: sequences and series. Each time, the collection of points was either finite or countable and the most important property of a point, in a sense, was its location in some coordinate or number system. }\li0\ri0\sa160\sb0\fi0\ql\par} {\f3 {\ltrch }\li0\ri0\sa160\sb0\fi0\ql\par} {\f3 {\ltrch Now will deal with points, or more precisely with sets of points, in a more abstract setting. Concepts such as addition and multiplication will not work anymore, and we will have to start, in a sense, at the beginning again: }\li0\ri0\sa160\sb0\fi0\ql\par} {\f3 {\ltrch }\li0\ri0\sa160\sb0\fi0\ql\par} {\f3 {\ltrch we need to define the basic objects we want to deal with, together with their most elementary properties; then we will develop a theory of those objects and called it }{\b\ltrch topology}{\ltrch .}\li0\ri0\sa160\sb0\fi0\ql\par} {\f3 {\ltrch }\li0\ri0\sa160\sb0\fi0\ql\par} } }<FlowDocument FontFamily="Microsoft Sans Serif" FontStyle="Normal" FontWeight="Normal" FontSize="26.6766666666667" Foreground="#FF000000" PagePadding="5,0,5,0" AllowDrop="True" NumberSubstitution.CultureSource="User" xmlns="http://schemas.microsoft.com/winfx/2006/xaml/presentation"><Paragraph Margin="0,0,0,10.67" LineHeight="Auto" FontFamily="Trebuchet MS" FontWeight="Bold" FontSize="18.6666666666667"><Run FontFamily="Trebuchet MS" FontSize="26.6766666666667">Topology</Run></Paragraph><Paragraph Margin="0,0,0,10.67" LineHeight="Auto" FontFamily="Trebuchet MS" FontSize="14.6666666666667"><Run FontSize="26.6766666666667" xml:space="preserve">In the previous chapters we dealt with collections of points: sequences and series. Each time, the collection of points was either finite or countable and the most important property of a point, in a sense, was its location in some coordinate or number system. </Run></Paragraph><Paragraph Margin="0,0,0,10.67" LineHeight="Auto" FontFamily="Trebuchet MS" FontSize="14.6666666666667"><Run FontSize="26.6766666666667" xml:space="preserve" /></Paragraph><Paragraph Margin="0,0,0,10.67" LineHeight="Auto" FontFamily="Trebuchet MS" FontSize="14.6666666666667"><Run FontSize="26.6766666666667" xml:space="preserve">Now will deal with points, or more precisely with sets of points, in a more abstract setting. Concepts such as addition and multiplication will not work anymore, and we will have to start, in a sense, at the beginning again: </Run></Paragraph><Paragraph Margin="0,0,0,10.67" LineHeight="Auto" FontFamily="Trebuchet MS" FontSize="14.6666666666667"><Run FontSize="26.6766666666667" xml:space="preserve" /></Paragraph><Paragraph Margin="0,0,0,10.67" LineHeight="Auto" FontFamily="Trebuchet MS" FontSize="14.6666666666667"><Run FontSize="26.6766666666667" xml:space="preserve">we need to define the basic objects we want to deal with, together with their most elementary properties; then we will develop a theory of those objects and called it </Run><Run FontWeight="Bold" FontSize="26.6766666666667">topology</Run><Run FontSize="26.6766666666667">.</Run></Paragraph><Paragraph Margin="0,0,0,10.67" LineHeight="Auto" FontFamily="Trebuchet MS" FontSize="14.6666666666667"><Run FontSize="26.6766666666667" xml:space="preserve" /></Paragraph></FlowDocument>2624719209{\rtf1\ansi\ansicpg1252\uc1\htmautsp\deff2{\fonttbl{\f0\fcharset0 Times New Roman;}{\f2\fcharset0 Arial;}{\f3\fcharset0 Trebuchet MS;}}{\colortbl\red0\green0\blue0;\red255\green255\blue255;\red237\green242\blue242;} {\*\listtable {\list\listtemplateid1\listhybrid {\listlevel\levelnfc0\levelnfcn0\leveljc0\leveljcn0\levelfollow0\levelstartat1\levelspace0\levelindent0{\leveltext\leveltemplateid5\'02\'00.;}{\levelnumbers\'01;}\fi-360\li720\lin720\jclisttab\tx720} {\listlevel\levelnfc0\levelnfcn0\leveljc0\leveljcn0\levelfollow0\levelstartat1\levelspace0\levelindent0{\leveltext\leveltemplateid6\'02\'01.;}{\levelnumbers\'01;}\fi-360\li1440\lin1440\jclisttab\tx1440} {\listlevel\levelnfc0\levelnfcn0\leveljc0\leveljcn0\levelfollow0\levelstartat1\levelspace0\levelindent0{\leveltext\leveltemplateid7\'02\'02.;}{\levelnumbers\'01;}\fi-360\li2160\lin2160\jclisttab\tx2160} {\listlevel\levelnfc0\levelnfcn0\leveljc0\leveljcn0\levelfollow0\levelstartat1\levelspace0\levelindent0{\leveltext\leveltemplateid8\'02\'03.;}{\levelnumbers\'01;}\fi-360\li2880\lin2880\jclisttab\tx2880} {\listlevel\levelnfc0\levelnfcn0\leveljc0\leveljcn0\levelfollow0\levelstartat1\levelspace0\levelindent0{\leveltext\leveltemplateid9\'02\'04.;}{\levelnumbers\'01;}\fi-360\li3600\lin3600\jclisttab\tx3600} {\listlevel\levelnfc0\levelnfcn0\leveljc0\leveljcn0\levelfollow0\levelstartat1\levelspace0\levelindent0{\leveltext\leveltemplateid10\'02\'05.;}{\levelnumbers\'01;}\fi-360\li4320\lin4320\jclisttab\tx4320} {\listlevel\levelnfc0\levelnfcn0\leveljc0\leveljcn0\levelfollow0\levelstartat1\levelspace0\levelindent0{\leveltext\leveltemplateid11\'02\'06.;}{\levelnumbers\'01;}\fi-360\li5040\lin5040\jclisttab\tx5040} {\listlevel\levelnfc0\levelnfcn0\leveljc0\leveljcn0\levelfollow0\levelstartat1\levelspace0\levelindent0{\leveltext\leveltemplateid12\'02\'07.;}{\levelnumbers\'01;}\fi-360\li5760\lin5760\jclisttab\tx5760} {\listlevel\levelnfc0\levelnfcn0\leveljc0\leveljcn0\levelfollow0\levelstartat1\levelspace0\levelindent0{\leveltext\leveltemplateid13\'02\'08.;}{\levelnumbers\'01;}\fi-360\li6480\lin6480\jclisttab\tx6480} {\listname ;}\listid1}} {\*\listoverridetable {\listoverride\listid1\listoverridecount0\ls1} } \loch\hich\dbch\pard\tx1700\tx3400\tx5100\plain\ltrpar\itap0{\lang1033\fs44\f2\cf0 \cf0\ql{\fs40\f3 {\b\ltrch Example:}{\ltrch Which of the following sets are open, closed, both, or neither ?}\li0\ri0\sa0\sb0\fi0\ql\par} {{\pntext 1.\tab}{\*\pn\pnlvlbody\pndec\pnstart1{\pntxta .}}{\fs40\ltrch The intervals (-3, 3), [4, 7], (-4, 5], (0,\~}{\ltrch {\*\shppict{\pict\picwgoal189\pichgoal130\pngblip 89504e470d0a1a0a0000000d494844520000000d000000090806000000e97aa66a000000017352474200aece1ce90000000467414d410000b18f0bfc6105000000097048597300000ec300000ec301c76fa8640000004b4944415428539d90490e00200803a9ffff335a620d315c602e942a8bc2ddadcbbab1454c02f0c69d1c8cd923f209451c56977f4f79ac97bb48579e18bd298af24ad29527661fc1a21e661b134d37ff022d646f0000000049454e44ae426082}}}{\fs40\ltrch ) and [0,\~}{\ltrch {\*\shppict{\pict\picwgoal189\pichgoal130\pngblip 89504e470d0a1a0a0000000d494844520000000d000000090806000000e97aa66a000000017352474200aece1ce90000000467414d410000b18f0bfc6105000000097048597300000ec300000ec301c76fa8640000004b4944415428539d90490e00200803a9ffff335a620d315c602e942a8bc2ddadcbbab1454c02f0c69d1c8cd923f209451c56977f4f79ac97bb48579e18bd298af24ad29527661fc1a21e661b134d37ff022d646f0000000049454e44ae426082}}}{\fs40\ltrch )}\li720\ri0\sa100\sb100\jclisttab\tx720\fi-360\ql\cbpat2\par} {\fs40 {\ltrch }\li720\ri0\sa100\sb100\fi0\ql\cbpat2\par} {\fs40 {\pntext 2.\tab}{\*\pn\pnlvlbody\pndec\pnstart1{\pntxta .}}{\ltrch The sets R (the whole real line) and 0 (the empty set)}\li720\ri0\sa100\sb100\jclisttab\tx720\fi-360\ql\cbpat2\par} {\fs40 {\ltrch }\li720\ri0\sa100\sb100\fi0\ql\cbpat2\par} {{\pntext 3.\tab}{\*\pn\pnlvlbody\pndec\pnstart1{\pntxta .}}{\fs40\ltrch The set \{1, 1/2, 1/3, 1/4, 1/5, ...\} and \{1, 1/2, 1/3, 1/4, ...\}\~}{\ltrch {\*\shppict{\pict\picwgoal180\pichgoal150\pngblip 89504e470d0a1a0a0000000d494844520000000c0000000a0806000000802cbffa000000017352474200aece1ce90000000467414d410000b18f0bfc6105000000097048597300000ec300000ec301c76fa8640000004649444154285363fcffff3f032980094a130d98181919095a81ac86741ba034d180740dc050023a11b73f40722035502ec4065c9ad01583004a3ca06b42570c0224461c030300eb361f0d4bd1aba10000000049454e44ae426082}}}{\fs40\ltrch \{\}\{0}\li720\ri0\sa100\sb100\jclisttab\tx720\fi-360\ql\cbpat2\par} {\fs40\f2 {\ltrch }\li0\ri0\sa0\sb0\fi0\ql\par} } }<FlowDocument FontFamily="Arial" FontStyle="Normal" FontWeight="Normal" FontSize="29.3433333333333" Foreground="#FF000000" PagePadding="5,0,5,0" AllowDrop="True" NumberSubstitution.CultureSource="User" xmlns="http://schemas.microsoft.com/winfx/2006/xaml/presentation"><Paragraph Margin="0,0,0,0" LineHeight="Auto" FontFamily="Times New Roman" FontSize="16"><Run FontFamily="Trebuchet MS" FontWeight="Bold" FontSize="26.6766666666667">Example:</Run><Run FontFamily="Trebuchet MS" FontSize="26.6766666666667" xml:space="preserve"> Which of the following sets are open, closed, both, or neither ?</Run></Paragraph><List MarkerStyle="Decimal" MarkerOffset="17" Margin="0,0,0,0" Padding="0,0,0,0" FontFamily="Trebuchet MS" FontSize="26.6766666666667"><ListItem Margin="48,0,0,0" Padding="0,0,0,0" BorderThickness="1,1,1,1" FontSize="14.6766666666667"><Paragraph Margin="0,6.67,0,6.67" LineHeight="Auto" FontSize="14.6666666666667" Background="#FFEDF2F2"><Run FontSize="26.6766666666667">The intervals (-3, 3), [4, 7], (-4, 5], (0, </Run><InlineUIContainer FontSize="14.6766666666667"><Image Stretch="Fill" Width="12.61" Height="8.64" InputMethod.IsInputMethodEnabled="True"><Image.Source><BitmapImage BaseUri="file:///C:/Users/wachs/AppData/Local/Temp/DyKnow/img30/eac83b4f-4eae-4100-9c4c-7c2fa98ccbd6" UriSource="file:///C:/Users/wachs/AppData/Local/Temp/DyKnow/img30/eac83b4f-4eae-4100-9c4c-7c2fa98ccbd6" CacheOption="OnLoad" /></Image.Source></Image></InlineUIContainer><Run FontSize="26.6766666666667">) and [0, </Run><InlineUIContainer FontSize="14.6766666666667"><Image Stretch="Fill" Width="12.61" Height="8.64" InputMethod.IsInputMethodEnabled="True"><Image.Source><BitmapImage BaseUri="file:///C:/Users/wachs/AppData/Local/Temp/DyKnow/img30/eac83b4f-4eae-4100-9c4c-7c2fa98ccbd6" UriSource="file:///C:/Users/wachs/AppData/Local/Temp/DyKnow/img30/eac83b4f-4eae-4100-9c4c-7c2fa98ccbd6" CacheOption="OnLoad" /></Image.Source></Image></InlineUIContainer><Run FontSize="26.6766666666667">)</Run></Paragraph><Paragraph Margin="0,6.67,0,6.67" LineHeight="Auto" FontSize="14.6666666666667" Background="#FFEDF2F2"><Run FontSize="26.6766666666667" xml:space="preserve" /></Paragraph></ListItem><ListItem Margin="48,0,0,0" Padding="0,0,0,0" BorderThickness="1,1,1,1" FontSize="14.6766666666667"><Paragraph Margin="0,6.67,0,6.67" LineHeight="Auto" FontSize="14.6666666666667" Background="#FFEDF2F2"><Run FontSize="26.6766666666667">The sets R (the whole real line) and 0 (the empty set)</Run></Paragraph><Paragraph Margin="0,6.67,0,6.67" LineHeight="Auto" FontSize="14.6666666666667" Background="#FFEDF2F2"><Run FontSize="26.6766666666667" xml:space="preserve" /></Paragraph></ListItem><ListItem Margin="48,0,0,0" Padding="0,0,0,0" BorderThickness="1,1,1,1" FontSize="14.6766666666667"><Paragraph Margin="0,6.67,0,6.67" LineHeight="Auto" FontSize="14.6666666666667" Background="#FFEDF2F2"><Run FontSize="26.6766666666667">The set {1, 1/2, 1/3, 1/4, 1/5, ...} and {1, 1/2, 1/3, 1/4, ...} </Run><InlineUIContainer FontSize="14.6766666666667"><Image Stretch="Fill" Width="12" Height="10" InputMethod.IsInputMethodEnabled="True"><Image.Source><BitmapImage BaseUri="file:///C:/Users/wachs/AppData/Local/Temp/DyKnow/img30/d42fdf07-15ea-452c-ba06-0d503deeec90" UriSource="file:///C:/Users/wachs/AppData/Local/Temp/DyKnow/img30/d42fdf07-15ea-452c-ba06-0d503deeec90" CacheOption="OnLoad" /></Image.Source></Image></InlineUIContainer><Run FontFamily="Trebuchet MS" FontSize="26.6766666666667">{}{0</Run></Paragraph></ListItem></List><Paragraph LineHeight="Auto"><Run FontSize="26.6766666666667" xml:space="preserve" /></Paragraph></FlowDocument>2624711775<FlowDocument FontFamily="Arial" FontStyle="Normal" FontWeight="Normal" FontSize="29.3433333333333" Foreground="#FF000000" PagePadding="5,0,5,0" AllowDrop="True" NumberSubstitution.CultureSource="User" xmlns="http://schemas.microsoft.com/winfx/2006/xaml/presentation"><Paragraph Margin="0,0,0,0" LineHeight="Auto" FontFamily="Times New Roman" FontSize="16"><Run FontFamily="Trebuchet MS" FontWeight="Bold" FontSize="26.6766666666667">Example:</Run><Run FontFamily="Trebuchet MS" FontSize="26.6766666666667" xml:space="preserve"> Which of the following sets are open, closed, both, or neither ?</Run></Paragraph><List MarkerStyle="Decimal" MarkerOffset="17" Margin="0,0,0,0" Padding="0,0,0,0" FontFamily="Trebuchet MS" FontSize="26.6766666666667"><ListItem Margin="48,0,0,0" Padding="0,0,0,0" BorderThickness="1,1,1,1" FontSize="14.6766666666667"><Paragraph Margin="0,6.67,0,6.67" LineHeight="Auto" FontSize="14.6666666666667" Background="#FFEDF2F2"><Run FontSize="26.6766666666667">The intervals (-3, 3), [4, 7], (-4, 5], (0, </Run><InlineUIContainer FontSize="14.6766666666667"><Image Stretch="Fill" Width="12.61" Height="8.64" InputMethod.IsInputMethodEnabled="True"><Image.Source><BitmapImage BaseUri="file:///C:/Users/wachs/AppData/Local/Temp/DyKnow/img30/eac83b4f-4eae-4100-9c4c-7c2fa98ccbd6" UriSource="file:///C:/Users/wachs/AppData/Local/Temp/DyKnow/img30/eac83b4f-4eae-4100-9c4c-7c2fa98ccbd6" CacheOption="OnLoad" /></Image.Source></Image></InlineUIContainer><Run FontSize="26.6766666666667">) and [0, </Run><InlineUIContainer FontSize="14.6766666666667"><Image Stretch="Fill" Width="12.61" Height="8.64" InputMethod.IsInputMethodEnabled="True"><Image.Source><BitmapImage BaseUri="file:///C:/Users/wachs/AppData/Local/Temp/DyKnow/img30/eac83b4f-4eae-4100-9c4c-7c2fa98ccbd6" UriSource="file:///C:/Users/wachs/AppData/Local/Temp/DyKnow/img30/eac83b4f-4eae-4100-9c4c-7c2fa98ccbd6" CacheOption="OnLoad" /></Image.Source></Image></InlineUIContainer><Run FontSize="26.6766666666667">)</Run></Paragraph><Paragraph Margin="0,6.67,0,6.67" LineHeight="Auto" FontSize="14.6666666666667" Background="#FFEDF2F2"><Run FontSize="26.6766666666667" xml:space="preserve" /></Paragraph></ListItem><ListItem Margin="48,0,0,0" Padding="0,0,0,0" BorderThickness="1,1,1,1" FontSize="14.6766666666667"><Paragraph Margin="0,6.67,0,6.67" LineHeight="Auto" FontSize="14.6666666666667" Background="#FFEDF2F2"><Run FontSize="26.6766666666667">The sets R (the whole real line) and 0 (the empty set)</Run></Paragraph><Paragraph Margin="0,6.67,0,6.67" LineHeight="Auto" FontSize="14.6666666666667" Background="#FFEDF2F2"><Run FontSize="26.6766666666667" xml:space="preserve" /></Paragraph></ListItem><ListItem Margin="48,0,0,0" Padding="0,0,0,0" BorderThickness="1,1,1,1" FontSize="14.6766666666667"><Paragraph Margin="0,6.67,0,6.67" LineHeight="Auto" FontSize="14.6666666666667" Background="#FFEDF2F2"><Run FontSize="26.6766666666667">The set {1, 1/2, 1/3, 1/4, 1/5, ...} and {1, 1/2, 1/3, 1/4, ...} </Run><InlineUIContainer FontSize="14.6766666666667"><Image Stretch="Fill" Width="12" Height="10" InputMethod.IsInputMethodEnabled="True"><Image.Source><BitmapImage BaseUri="file:///C:/Users/wachs/AppData/Local/Temp/DyKnow/img30/d42fdf07-15ea-452c-ba06-0d503deeec90" UriSource="file:///C:/Users/wachs/AppData/Local/Temp/DyKnow/img30/d42fdf07-15ea-452c-ba06-0d503deeec90" CacheOption="OnLoad" /></Image.Source></Image></InlineUIContainer><Run FontFamily="Trebuchet MS" FontSize="26.6766666666667">{}{0</Run></Paragraph></ListItem></List><Paragraph LineHeight="Auto"><Run FontSize="26.6766666666667" xml:space="preserve" /></Paragraph></FlowDocument>{\rtf1\ansi\ansicpg1252\uc1\htmautsp\deff2{\fonttbl{\f0\fcharset0 Times New Roman;}{\f2\fcharset0 Microsoft Sans Serif;}{\f3\fcharset0 Trebuchet MS;}}{\colortbl\red0\green0\blue0;\red255\green255\blue255;\red64\green38\blue64;}\loch\hich\dbch\pard\tx1700\tx3400\tx5100\plain\ltrpar\itap0{\lang1033\fs40\f3\cf0 \cf0\ql{\f3\b {\cf2\ltrch Definition 5.1.1: Open and Closed Sets}\li0\ri0\sa100\sb0\brdrl\brdrnone\brdrcf0\brdrt\brdrnone\brdrcf0\brdrr\brdrnone\brdrcf0\brdrb\brdrs\brdrw15\brdrcf0\brdrcf0\brsp0\fi0\ql\cbpat1\par} {\fs22\f3\b {\fs40\ltrch A set\~U\~}{\ltrch {\*\shppict{\pict\picwgoal189\pichgoal150\pngblip 89504e470d0a1a0a0000000d494844520000000d0000000a08060000006feed4c4000000017352474200aece1ce90000000467414d410000b18f0bfc6105000000097048597300000ec300000ec301c76fa8640000004049444154285363fcffff3f03a980094a9304506c626464c46b2d502d2388866b02698009120260e791a20104c8f213f99a404e231408c880b2d0231e30300000247923015c59e70a0000000049454e44ae426082}}}{\fs40\ltrch \~R\~is called\~open, if for each\~}{\i\fs40\ltrch x\~}{\ltrch {\*\shppict{\pict\picwgoal189\pichgoal150\pngblip 89504e470d0a1a0a0000000d494844520000000d0000000a08060000006feed4c4000000017352474200aece1ce90000000467414d410000b18f0bfc6105000000097048597300000ec300000ec301c76fa8640000004d494441542853958ec112001008057bfeff9f23a3866ad05e3858fbc0cc54a5adb344280108e9f106eb3a392411fc830c9bf72b0856ca6629feb343fa2dd93c116eb51d2b2999e81704e90d510760d530f9f237cd3b0000000049454e44ae426082}}}{\i\fs40\ltrch \~U}{\fs40\ltrch \~there exists and\~}{\ltrch {\*\shppict{\pict\picwgoal150\pichgoal150\pngblip 89504e470d0a1a0a0000000d494844520000000a0000000a08060000008d32cfbd000000017352474200aece1ce90000000467414d410000b18f0bfc6105000000097048597300000ec300000ec301c76fa8640000004c4944415428538d8f010ac0200c031bffffe7ba809176b56e0752b121877077fbc358f393632380fdf8ecc1991a19e0e1520151d43110ef452dedbb7107bb8048eaa40a1f22a551b4ea3b6613136033f960e900480000000049454e44ae426082}}}{\i\fs40\ltrch \~> 0}{\fs40\ltrch \~such that the interval\~}{\i\fs40\ltrch ( x -\~}{\ltrch {\*\shppict{\pict\picwgoal150\pichgoal150\pngblip 89504e470d0a1a0a0000000d494844520000000a0000000a08060000008d32cfbd000000017352474200aece1ce90000000467414d410000b18f0bfc6105000000097048597300000ec300000ec301c76fa8640000004c4944415428538d8f010ac0200c031bffffe7ba809176b56e0752b121877077fbc358f393632380fdf8ecc1991a19e0e1520151d43110ef452dedbb7107bb8048eaa40a1f22a551b4ea3b6613136033f960e900480000000049454e44ae426082}}}{\i\fs40\ltrch , x +\~}{\ltrch {\*\shppict{\pict\picwgoal150\pichgoal150\pngblip 89504e470d0a1a0a0000000d494844520000000a0000000a08060000008d32cfbd000000017352474200aece1ce90000000467414d410000b18f0bfc6105000000097048597300000ec300000ec301c76fa8640000004c4944415428538d8f010ac0200c031bffffe7ba809176b56e0752b121877077fbc358f393632380fdf8ecc1991a19e0e1520151d43110ef452dedbb7107bb8048eaa40a1f22a551b4ea3b6613136033f960e900480000000049454e44ae426082}}}{\i\fs40\ltrch )}{\fs40\ltrch \~is contained in\~U. Such an interval is often called an\~}{\ltrch {\*\shppict{\pict\picwgoal150\pichgoal150\pngblip 89504e470d0a1a0a0000000d494844520000000a0000000a08060000008d32cfbd000000017352474200aece1ce90000000467414d410000b18f0bfc6105000000097048597300000ec300000ec301c76fa8640000004c4944415428538d8f010ac0200c031bffffe7ba809176b56e0752b121877077fbc358f393632380fdf8ecc1991a19e0e1520151d43110ef452dedbb7107bb8048eaa40a1f22a551b4ea3b6613136033f960e900480000000049454e44ae426082}}}{\fs40\ltrch \~-neighborhood\~of\~}{\i\fs40\ltrch x}{\fs40\ltrch , or simply a neighborhood of\~}{\i\fs40\ltrch x}{\fs40\ltrch .}\li0\ri0\sa0\sb0\fi0\ql\cbpat1\par} {\f3\b {\ltrch A set\~F\~is called\~closed\~if the complement of\~F,\~R\~\\\~F, is open.}\li0\ri0\sa100\sb100\fi0\ql\cbpat1\par} } }<FlowDocument FontFamily="Microsoft Sans Serif" FontStyle="Normal" FontWeight="Normal" FontSize="26.6766666666667" Foreground="#FF000000" PagePadding="5,0,5,0" AllowDrop="True" NumberSubstitution.CultureSource="User" xmlns="http://schemas.microsoft.com/winfx/2006/xaml/presentation"><Paragraph Margin="0,0,0,6.67" BorderThickness="0,0,0,1" BorderBrush="#FF000000" LineHeight="Auto" FontFamily="Trebuchet MS" FontWeight="Bold" FontSize="24" Background="#FFFFFFFF"><Run FontFamily="Trebuchet MS" FontSize="26.6766666666667" Foreground="#FF402640">Definition 5.1.1: Open and Closed Sets</Run></Paragraph><Paragraph Margin="0,0,0,0" LineHeight="Auto" FontFamily="Trebuchet MS" FontWeight="Bold" FontSize="14.6666666666667" Background="#FFFFFFFF"><Run FontSize="26.6766666666667">A set U </Run><InlineUIContainer FontSize="14.6766666666667"><Image Stretch="Fill" Width="12.61" Height="10" InputMethod.IsInputMethodEnabled="True"><Image.Source><BitmapImage BaseUri="file:///C:/Users/wachs/AppData/Local/Temp/DyKnow/img30/84f739cb-268f-4bb2-9582-1a6fefea24a3" UriSource="file:///C:/Users/wachs/AppData/Local/Temp/DyKnow/img30/84f739cb-268f-4bb2-9582-1a6fefea24a3" CacheOption="OnLoad" /></Image.Source></Image></InlineUIContainer><Run FontSize="26.6766666666667"> R is called open, if for each </Run><Run FontStyle="Italic" FontSize="26.6766666666667">x </Run><InlineUIContainer FontSize="14.6766666666667"><Image Stretch="Fill" Width="12.61" Height="10" InputMethod.IsInputMethodEnabled="True"><Image.Source><BitmapImage BaseUri="file:///C:/Users/wachs/AppData/Local/Temp/DyKnow/img30/2d1283a1-866f-4432-86dc-b343e7a26d38" UriSource="file:///C:/Users/wachs/AppData/Local/Temp/DyKnow/img30/2d1283a1-866f-4432-86dc-b343e7a26d38" CacheOption="OnLoad" /></Image.Source></Image></InlineUIContainer><Run FontStyle="Italic" FontSize="26.6766666666667"> U</Run><Run FontSize="26.6766666666667"> there exists and </Run><InlineUIContainer FontSize="14.6766666666667"><Image Stretch="Fill" Width="10" Height="10" InputMethod.IsInputMethodEnabled="True"><Image.Source><BitmapImage BaseUri="file:///C:/Users/wachs/AppData/Local/Temp/DyKnow/img30/b62b6b58-2ad5-4419-9118-7e41cac63498" UriSource="file:///C:/Users/wachs/AppData/Local/Temp/DyKnow/img30/b62b6b58-2ad5-4419-9118-7e41cac63498" CacheOption="OnLoad" /></Image.Source></Image></InlineUIContainer><Run FontStyle="Italic" FontSize="26.6766666666667"> &gt; 0</Run><Run FontSize="26.6766666666667"> such that the interval </Run><Run FontStyle="Italic" FontSize="26.6766666666667">( x - </Run><InlineUIContainer FontSize="14.6766666666667"><Image Stretch="Fill" Width="10" Height="10" InputMethod.IsInputMethodEnabled="True"><Image.Source><BitmapImage BaseUri="file:///C:/Users/wachs/AppData/Local/Temp/DyKnow/img30/b62b6b58-2ad5-4419-9118-7e41cac63498" UriSource="file:///C:/Users/wachs/AppData/Local/Temp/DyKnow/img30/b62b6b58-2ad5-4419-9118-7e41cac63498" CacheOption="OnLoad" /></Image.Source></Image></InlineUIContainer><Run FontStyle="Italic" FontSize="26.6766666666667">, x + </Run><InlineUIContainer FontSize="14.6766666666667"><Image Stretch="Fill" Width="10" Height="10" InputMethod.IsInputMethodEnabled="True"><Image.Source><BitmapImage BaseUri="file:///C:/Users/wachs/AppData/Local/Temp/DyKnow/img30/b62b6b58-2ad5-4419-9118-7e41cac63498" UriSource="file:///C:/Users/wachs/AppData/Local/Temp/DyKnow/img30/b62b6b58-2ad5-4419-9118-7e41cac63498" CacheOption="OnLoad" /></Image.Source></Image></InlineUIContainer><Run FontStyle="Italic" FontSize="26.6766666666667">)</Run><Run FontSize="26.6766666666667"> is contained in U. Such an interval is often called an </Run><InlineUIContainer FontSize="14.6766666666667"><Image Stretch="Fill" Width="10" Height="10" InputMethod.IsInputMethodEnabled="True"><Image.Source><BitmapImage BaseUri="file:///C:/Users/wachs/AppData/Local/Temp/DyKnow/img30/b62b6b58-2ad5-4419-9118-7e41cac63498" UriSource="file:///C:/Users/wachs/AppData/Local/Temp/DyKnow/img30/b62b6b58-2ad5-4419-9118-7e41cac63498" CacheOption="OnLoad" /></Image.Source></Image></InlineUIContainer><Run FontSize="26.6766666666667"> -neighborhood of </Run><Run FontStyle="Italic" FontSize="26.6766666666667">x</Run><Run FontSize="26.6766666666667">, or simply a neighborhood of </Run><Run FontStyle="Italic" FontSize="26.6766666666667">x</Run><Run FontSize="26.6766666666667">.</Run></Paragraph><Paragraph Margin="0,6.67,0,6.67" LineHeight="Auto" FontFamily="Trebuchet MS" FontWeight="Bold" FontSize="14.6666666666667" Background="#FFFFFFFF"><Run FontSize="26.6766666666667">A set F is called closed if the complement of F, R \ F, is open.</Run></Paragraph></FlowDocument>2624714589<FlowDocument FontFamily="Arial" FontStyle="Normal" FontWeight="Normal" FontSize="29.3433333333333" Foreground="#FF000000" PagePadding="5,0,5,0" AllowDrop="True" NumberSubstitution.CultureSource="User" xmlns="http://schemas.microsoft.com/winfx/2006/xaml/presentation"><Paragraph Margin="0,0,0,0" LineHeight="Auto" FontFamily="Times New Roman" FontSize="16"><Run FontFamily="Trebuchet MS" FontWeight="Bold" FontSize="26.6766666666667">Example:</Run><Run FontFamily="Trebuchet MS" FontSize="26.6766666666667" xml:space="preserve"> Which of the following sets are open, closed, both, or neither ?</Run></Paragraph><List MarkerStyle="Decimal" MarkerOffset="17" Margin="0,0,0,0" Padding="0,0,0,0" FontFamily="Trebuchet MS" FontSize="26.6766666666667"><ListItem Margin="48,0,0,0" Padding="0,0,0,0" BorderThickness="1,1,1,1" FontSize="14.6766666666667"><Paragraph Margin="0,6.67,0,6.67" LineHeight="Auto" FontSize="14.6666666666667" Background="#FFEDF2F2"><Run FontSize="26.6766666666667">The intervals (-3, 3), [4, 7], (-4, 5], (0, </Run><InlineUIContainer FontSize="14.6766666666667"><Image Stretch="Fill" Width="12.61" Height="8.64" InputMethod.IsInputMethodEnabled="True"><Image.Source><BitmapImage BaseUri="file:///C:/Users/wachs/AppData/Local/Temp/DyKnow/img30/eac83b4f-4eae-4100-9c4c-7c2fa98ccbd6" UriSource="file:///C:/Users/wachs/AppData/Local/Temp/DyKnow/img30/eac83b4f-4eae-4100-9c4c-7c2fa98ccbd6" CacheOption="OnLoad" /></Image.Source></Image></InlineUIContainer><Run FontSize="26.6766666666667">) and [0, </Run><InlineUIContainer FontSize="14.6766666666667"><Image Stretch="Fill" Width="12.61" Height="8.64" InputMethod.IsInputMethodEnabled="True"><Image.Source><BitmapImage BaseUri="file:///C:/Users/wachs/AppData/Local/Temp/DyKnow/img30/eac83b4f-4eae-4100-9c4c-7c2fa98ccbd6" UriSource="file:///C:/Users/wachs/AppData/Local/Temp/DyKnow/img30/eac83b4f-4eae-4100-9c4c-7c2fa98ccbd6" CacheOption="OnLoad" /></Image.Source></Image></InlineUIContainer><Run FontSize="26.6766666666667">)</Run></Paragraph><Paragraph Margin="0,6.67,0,6.67" LineHeight="Auto" FontSize="14.6666666666667" Background="#FFEDF2F2"><Run FontSize="26.6766666666667" xml:space="preserve" /></Paragraph></ListItem><ListItem Margin="48,0,0,0" Padding="0,0,0,0" BorderThickness="1,1,1,1" FontSize="14.6766666666667"><Paragraph Margin="0,6.67,0,6.67" LineHeight="Auto" FontSize="14.6666666666667" Background="#FFEDF2F2"><Run FontSize="26.6766666666667">The sets R (the whole real line) and 0 (the empty set)</Run></Paragraph><Paragraph Margin="0,6.67,0,6.67" LineHeight="Auto" FontSize="14.6666666666667" Background="#FFEDF2F2"><Run FontSize="26.6766666666667" xml:space="preserve" /></Paragraph></ListItem><ListItem Margin="48,0,0,0" Padding="0,0,0,0" BorderThickness="1,1,1,1" FontSize="14.6766666666667"><Paragraph Margin="0,6.67,0,6.67" LineHeight="Auto" FontSize="14.6666666666667" Background="#FFEDF2F2"><Run FontSize="26.6766666666667">The set {1, 1/2, 1/3, 1/4, 1/5, ...} and {1, 1/2, 1/3, 1/4, ...} </Run><InlineUIContainer FontSize="14.6766666666667"><Image Stretch="Fill" Width="12" Height="10" InputMethod.IsInputMethodEnabled="True"><Image.Source><BitmapImage BaseUri="file:///C:/Users/wachs/AppData/Local/Temp/DyKnow/img30/d42fdf07-15ea-452c-ba06-0d503deeec90" UriSource="file:///C:/Users/wachs/AppData/Local/Temp/DyKnow/img30/d42fdf07-15ea-452c-ba06-0d503deeec90" CacheOption="OnLoad" /></Image.Source></Image></InlineUIContainer><Run FontFamily="Trebuchet MS" FontSize="26.6766666666667">{}{0</Run></Paragraph></ListItem></List><Paragraph LineHeight="Auto"><Run FontSize="26.6766666666667" xml:space="preserve" /></Paragraph></FlowDocument><FlowDocument FontFamily="Microsoft Sans Serif" FontStyle="Normal" FontWeight="Normal" FontSize="26.6766666666667" Foreground="#FF000000" PagePadding="5,0,5,0" AllowDrop="True" NumberSubstitution.CultureSource="User" xmlns="http://schemas.microsoft.com/winfx/2006/xaml/presentation"><Paragraph Margin="0,0,0,6.67" BorderThickness="0,0,0,1" BorderBrush="#FF000000" LineHeight="Auto" FontFamily="Trebuchet MS" FontWeight="Bold" FontSize="24" Background="#FFFFFFFF"><Run FontFamily="Trebuchet MS" FontSize="26.6766666666667" Foreground="#FF402640">Definition 5.1.1: Open and Closed Sets</Run></Paragraph><Paragraph Margin="0,0,0,0" LineHeight="Auto" FontFamily="Trebuchet MS" FontWeight="Bold" FontSize="14.6666666666667" Background="#FFFFFFFF"><Run FontSize="26.6766666666667">A set U </Run><InlineUIContainer FontSize="14.6766666666667"><Image Stretch="Fill" Width="12.61" Height="10" InputMethod.IsInputMethodEnabled="True"><Image.Source><BitmapImage BaseUri="file:///C:/Users/wachs/AppData/Local/Temp/DyKnow/img30/84f739cb-268f-4bb2-9582-1a6fefea24a3" UriSource="file:///C:/Users/wachs/AppData/Local/Temp/DyKnow/img30/84f739cb-268f-4bb2-9582-1a6fefea24a3" CacheOption="OnLoad" /></Image.Source></Image></InlineUIContainer><Run FontSize="26.6766666666667"> R is called open, if for each </Run><Run FontStyle="Italic" FontSize="26.6766666666667">x </Run><InlineUIContainer FontSize="14.6766666666667"><Image Stretch="Fill" Width="12.61" Height="10" InputMethod.IsInputMethodEnabled="True"><Image.Source><BitmapImage BaseUri="file:///C:/Users/wachs/AppData/Local/Temp/DyKnow/img30/2d1283a1-866f-4432-86dc-b343e7a26d38" UriSource="file:///C:/Users/wachs/AppData/Local/Temp/DyKnow/img30/2d1283a1-866f-4432-86dc-b343e7a26d38" CacheOption="OnLoad" /></Image.Source></Image></InlineUIContainer><Run FontStyle="Italic" FontSize="26.6766666666667"> U</Run><Run FontSize="26.6766666666667"> there exists and </Run><InlineUIContainer FontSize="14.6766666666667"><Image Stretch="Fill" Width="10" Height="10" InputMethod.IsInputMethodEnabled="True"><Image.Source><BitmapImage BaseUri="file:///C:/Users/wachs/AppData/Local/Temp/DyKnow/img30/b62b6b58-2ad5-4419-9118-7e41cac63498" UriSource="file:///C:/Users/wachs/AppData/Local/Temp/DyKnow/img30/b62b6b58-2ad5-4419-9118-7e41cac63498" CacheOption="OnLoad" /></Image.Source></Image></InlineUIContainer><Run FontStyle="Italic" FontSize="26.6766666666667"> &gt; 0</Run><Run FontSize="26.6766666666667"> such that the interval </Run><Run FontStyle="Italic" FontSize="26.6766666666667">( x - </Run><InlineUIContainer FontSize="14.6766666666667"><Image Stretch="Fill" Width="10" Height="10" InputMethod.IsInputMethodEnabled="True"><Image.Source><BitmapImage BaseUri="file:///C:/Users/wachs/AppData/Local/Temp/DyKnow/img30/b62b6b58-2ad5-4419-9118-7e41cac63498" UriSource="file:///C:/Users/wachs/AppData/Local/Temp/DyKnow/img30/b62b6b58-2ad5-4419-9118-7e41cac63498" CacheOption="OnLoad" /></Image.Source></Image></InlineUIContainer><Run FontStyle="Italic" FontSize="26.6766666666667">, x + </Run><InlineUIContainer FontSize="14.6766666666667"><Image Stretch="Fill" Width="10" Height="10" InputMethod.IsInputMethodEnabled="True"><Image.Source><BitmapImage BaseUri="file:///C:/Users/wachs/AppData/Local/Temp/DyKnow/img30/b62b6b58-2ad5-4419-9118-7e41cac63498" UriSource="file:///C:/Users/wachs/AppData/Local/Temp/DyKnow/img30/b62b6b58-2ad5-4419-9118-7e41cac63498" CacheOption="OnLoad" /></Image.Source></Image></InlineUIContainer><Run FontStyle="Italic" FontSize="26.6766666666667">)</Run><Run FontSize="26.6766666666667"> is contained in U. Such an interval is often called an </Run><InlineUIContainer FontSize="14.6766666666667"><Image Stretch="Fill" Width="10" Height="10" InputMethod.IsInputMethodEnabled="True"><Image.Source><BitmapImage BaseUri="file:///C:/Users/wachs/AppData/Local/Temp/DyKnow/img30/b62b6b58-2ad5-4419-9118-7e41cac63498" UriSource="file:///C:/Users/wachs/AppData/Local/Temp/DyKnow/img30/b62b6b58-2ad5-4419-9118-7e41cac63498" CacheOption="OnLoad" /></Image.Source></Image></InlineUIContainer><Run FontSize="26.6766666666667"> -neighborhood of </Run><Run FontStyle="Italic" FontSize="26.6766666666667">x</Run><Run FontSize="26.6766666666667">, or simply a neighborhood of </Run><Run FontStyle="Italic" FontSize="26.6766666666667">x</Run><Run FontSize="26.6766666666667">.</Run></Paragraph><Paragraph Margin="0,6.67,0,6.67" LineHeight="Auto" FontFamily="Trebuchet MS" FontWeight="Bold" FontSize="14.6666666666667" Background="#FFFFFFFF"><Run FontSize="26.6766666666667">A set F is called closed if the complement of F, R \ F, is open.</Run></Paragraph></FlowDocument><FlowDocument FontFamily="Microsoft Sans Serif" FontStyle="Normal" FontWeight="Normal" FontSize="26.6766666666667" Foreground="#FF000000" PagePadding="5,0,5,0" AllowDrop="True" NumberSubstitution.CultureSource="User" xmlns="http://schemas.microsoft.com/winfx/2006/xaml/presentation"><Paragraph Margin="0,0,0,6.67" BorderThickness="0,0,0,1" BorderBrush="#FF000000" LineHeight="Auto" FontFamily="Trebuchet MS" FontWeight="Bold" FontSize="24" Background="#FFFFFFFF"><Run FontFamily="Trebuchet MS" FontSize="26.6766666666667" Foreground="#FF402640">Definition 5.1.1: Open and Closed Sets</Run></Paragraph><Paragraph Margin="0,0,0,0" LineHeight="Auto" FontFamily="Trebuchet MS" FontWeight="Bold" FontSize="14.6666666666667" Background="#FFFFFFFF"><Run FontSize="26.6766666666667">A set U </Run><InlineUIContainer FontSize="14.6766666666667"><Image Stretch="Fill" Width="12.61" Height="10" InputMethod.IsInputMethodEnabled="True"><Image.Source><BitmapImage BaseUri="file:///C:/Users/wachs/AppData/Local/Temp/DyKnow/img30/84f739cb-268f-4bb2-9582-1a6fefea24a3" UriSource="file:///C:/Users/wachs/AppData/Local/Temp/DyKnow/img30/84f739cb-268f-4bb2-9582-1a6fefea24a3" CacheOption="OnLoad" /></Image.Source></Image></InlineUIContainer><Run FontSize="26.6766666666667"> R is called open, if for each </Run><Run FontStyle="Italic" FontSize="26.6766666666667">x </Run><InlineUIContainer FontSize="14.6766666666667"><Image Stretch="Fill" Width="12.61" Height="10" InputMethod.IsInputMethodEnabled="True"><Image.Source><BitmapImage BaseUri="file:///C:/Users/wachs/AppData/Local/Temp/DyKnow/img30/2d1283a1-866f-4432-86dc-b343e7a26d38" UriSource="file:///C:/Users/wachs/AppData/Local/Temp/DyKnow/img30/2d1283a1-866f-4432-86dc-b343e7a26d38" CacheOption="OnLoad" /></Image.Source></Image></InlineUIContainer><Run FontStyle="Italic" FontSize="26.6766666666667"> U</Run><Run FontSize="26.6766666666667"> there exists and </Run><InlineUIContainer FontSize="14.6766666666667"><Image Stretch="Fill" Width="10" Height="10" InputMethod.IsInputMethodEnabled="True"><Image.Source><BitmapImage BaseUri="file:///C:/Users/wachs/AppData/Local/Temp/DyKnow/img30/b62b6b58-2ad5-4419-9118-7e41cac63498" UriSource="file:///C:/Users/wachs/AppData/Local/Temp/DyKnow/img30/b62b6b58-2ad5-4419-9118-7e41cac63498" CacheOption="OnLoad" /></Image.Source></Image></InlineUIContainer><Run FontStyle="Italic" FontSize="26.6766666666667"> &gt; 0</Run><Run FontSize="26.6766666666667"> such that the interval </Run><Run FontStyle="Italic" FontSize="26.6766666666667">( x - </Run><InlineUIContainer FontSize="14.6766666666667"><Image Stretch="Fill" Width="10" Height="10" InputMethod.IsInputMethodEnabled="True"><Image.Source><BitmapImage BaseUri="file:///C:/Users/wachs/AppData/Local/Temp/DyKnow/img30/b62b6b58-2ad5-4419-9118-7e41cac63498" UriSource="file:///C:/Users/wachs/AppData/Local/Temp/DyKnow/img30/b62b6b58-2ad5-4419-9118-7e41cac63498" CacheOption="OnLoad" /></Image.Source></Image></InlineUIContainer><Run FontStyle="Italic" FontSize="26.6766666666667">, x + </Run><InlineUIContainer FontSize="14.6766666666667"><Image Stretch="Fill" Width="10" Height="10" InputMethod.IsInputMethodEnabled="True"><Image.Source><BitmapImage BaseUri="file:///C:/Users/wachs/AppData/Local/Temp/DyKnow/img30/b62b6b58-2ad5-4419-9118-7e41cac63498" UriSource="file:///C:/Users/wachs/AppData/Local/Temp/DyKnow/img30/b62b6b58-2ad5-4419-9118-7e41cac63498" CacheOption="OnLoad" /></Image.Source></Image></InlineUIContainer><Run FontStyle="Italic" FontSize="26.6766666666667">)</Run><Run FontSize="26.6766666666667"> is contained in U. Such an interval is often called an </Run><InlineUIContainer FontSize="14.6766666666667"><Image Stretch="Fill" Width="10" Height="10" InputMethod.IsInputMethodEnabled="True"><Image.Source><BitmapImage BaseUri="file:///C:/Users/wachs/AppData/Local/Temp/DyKnow/img30/b62b6b58-2ad5-4419-9118-7e41cac63498" UriSource="file:///C:/Users/wachs/AppData/Local/Temp/DyKnow/img30/b62b6b58-2ad5-4419-9118-7e41cac63498" CacheOption="OnLoad" /></Image.Source></Image></InlineUIContainer><Run FontSize="26.6766666666667"> -neighborhood of </Run><Run FontStyle="Italic" FontSize="26.6766666666667">x</Run><Run FontSize="26.6766666666667">, or simply a neighborhood of </Run><Run FontStyle="Italic" FontSize="26.6766666666667">x</Run><Run FontSize="26.6766666666667">.</Run></Paragraph><Paragraph Margin="0,0,0,0" LineHeight="Auto" FontFamily="Trebuchet MS" FontWeight="Bold" FontSize="14.6666666666667" Background="#FFFFFFFF"><Run FontSize="26.6766666666667" xml:space="preserve" /></Paragraph><Paragraph Margin="0,6.67,0,6.67" LineHeight="Auto" FontFamily="Trebuchet MS" FontWeight="Bold" FontSize="14.6666666666667" Background="#FFFFFFFF"><Run FontSize="26.6766666666667">A set F is called closed if the complement of F, R \ F, is open.</Run></Paragraph></FlowDocument><FlowDocument FontFamily="Arial" FontStyle="Normal" FontWeight="Normal" FontSize="29.3433333333333" Foreground="#FF000000" PagePadding="5,0,5,0" AllowDrop="True" NumberSubstitution.CultureSource="User" xmlns="http://schemas.microsoft.com/winfx/2006/xaml/presentation"><Paragraph Margin="0,0,0,0" LineHeight="Auto" FontFamily="Times New Roman" FontSize="16"><Run FontFamily="Trebuchet MS" FontWeight="Bold" FontSize="26.6766666666667">Example:</Run><Run FontFamily="Trebuchet MS" FontSize="26.6766666666667" xml:space="preserve"> Which of the following sets are open, closed, both, or neither ?</Run></Paragraph><List MarkerStyle="Decimal" MarkerOffset="17" Margin="0,0,0,0" Padding="0,0,0,0" FontFamily="Trebuchet MS" FontSize="26.6766666666667"><ListItem Margin="48,0,0,0" Padding="0,0,0,0" BorderThickness="1,1,1,1" FontSize="14.6766666666667"><Paragraph Margin="0,6.67,0,6.67" LineHeight="Auto" FontSize="14.6666666666667" Background="#FFEDF2F2"><Run FontSize="26.6766666666667">The intervals (-3, 3), [4, 7], (-4, 5], (0, </Run><InlineUIContainer FontSize="14.6766666666667"><Image Stretch="Fill" Width="12.61" Height="8.64" InputMethod.IsInputMethodEnabled="True"><Image.Source><BitmapImage BaseUri="file:///C:/Users/wachs/AppData/Local/Temp/DyKnow/img30/eac83b4f-4eae-4100-9c4c-7c2fa98ccbd6" UriSource="file:///C:/Users/wachs/AppData/Local/Temp/DyKnow/img30/eac83b4f-4eae-4100-9c4c-7c2fa98ccbd6" CacheOption="OnLoad" /></Image.Source></Image></InlineUIContainer><Run FontSize="26.6766666666667">) and [0, </Run><InlineUIContainer FontSize="14.6766666666667"><Image Stretch="Fill" Width="12.61" Height="8.64" InputMethod.IsInputMethodEnabled="True"><Image.Source><BitmapImage BaseUri="file:///C:/Users/wachs/AppData/Local/Temp/DyKnow/img30/eac83b4f-4eae-4100-9c4c-7c2fa98ccbd6" UriSource="file:///C:/Users/wachs/AppData/Local/Temp/DyKnow/img30/eac83b4f-4eae-4100-9c4c-7c2fa98ccbd6" CacheOption="OnLoad" /></Image.Source></Image></InlineUIContainer><Run FontSize="26.6766666666667">)</Run></Paragraph><Paragraph Margin="0,6.67,0,6.67" LineHeight="Auto" FontSize="14.6666666666667" Background="#FFEDF2F2"><Run FontSize="26.6766666666667" xml:space="preserve" /></Paragraph></ListItem><ListItem Margin="48,0,0,0" Padding="0,0,0,0" BorderThickness="1,1,1,1" FontSize="14.6766666666667"><Paragraph Margin="0,6.67,0,6.67" LineHeight="Auto" FontSize="14.6666666666667" Background="#FFEDF2F2"><Run FontSize="26.6766666666667">The sets R (the whole real line) and 0 (the empty set)</Run></Paragraph><Paragraph Margin="0,6.67,0,6.67" LineHeight="Auto" FontSize="14.6666666666667" Background="#FFEDF2F2"><Run FontSize="26.6766666666667" xml:space="preserve" /></Paragraph></ListItem><ListItem Margin="48,0,0,0" Padding="0,0,0,0" BorderThickness="1,1,1,1" FontSize="14.6766666666667"><Paragraph Margin="0,6.67,0,6.67" LineHeight="Auto" FontSize="14.6666666666667" Background="#FFEDF2F2"><Run FontSize="26.6766666666667">The set {1, 1/2, 1/3, 1/4, 1/5, ...} and {1, 1/2, 1/3, 1/4, ...} </Run><InlineUIContainer FontSize="14.6766666666667"><Image Stretch="Fill" Width="12" Height="10" InputMethod.IsInputMethodEnabled="True"><Image.Source><BitmapImage BaseUri="file:///C:/Users/wachs/AppData/Local/Temp/DyKnow/img30/d42fdf07-15ea-452c-ba06-0d503deeec90" UriSource="file:///C:/Users/wachs/AppData/Local/Temp/DyKnow/img30/d42fdf07-15ea-452c-ba06-0d503deeec90" CacheOption="OnLoad" /></Image.Source></Image></InlineUIContainer><Run FontFamily="Trebuchet MS" FontSize="26.6766666666667">{}{0</Run></Paragraph></ListItem></List><Paragraph LineHeight="Auto"><Run FontSize="26.6766666666667" xml:space="preserve" /></Paragraph></FlowDocument><FlowDocument FontFamily="Microsoft Sans Serif" FontStyle="Normal" FontWeight="Normal" FontSize="26.6766666666667" Foreground="#FF000000" PagePadding="5,0,5,0" AllowDrop="True" NumberSubstitution.CultureSource="User" xmlns="http://schemas.microsoft.com/winfx/2006/xaml/presentation" xmlns:x="http://schemas.microsoft.com/winfx/2006/xaml"><Paragraph Margin="0,0,0,6.67" BorderThickness="0,0,0,1" BorderBrush="#FF000000" LineHeight="Auto" FontFamily="Trebuchet MS" FontWeight="Bold" FontSize="24" Background="#FFFFFFFF"><Run FontFamily="Trebuchet MS" FontSize="26.6766666666667" Foreground="#FF402640">Definition: Open and Closed Sets</Run></Paragraph><Paragraph Margin="0,0,0,0" LineHeight="Auto" FontFamily="Trebuchet MS" FontWeight="Bold" FontSize="14.6666666666667" Background="#FFFFFFFF"><Run FontSize="26.6766666666667">A set U </Run><InlineUIContainer FontSize="14.6766666666667"><Image Stretch="Fill" Width="12.61" Height="10" InputMethod.IsInputMethodEnabled="True"><Image.Source><BitmapImage BaseUri="file:///C:/Users/wachs/AppData/Local/Temp/DyKnow/img30/84f739cb-268f-4bb2-9582-1a6fefea24a3" UriSource="file:///C:/Users/wachs/AppData/Local/Temp/DyKnow/img30/84f739cb-268f-4bb2-9582-1a6fefea24a3" CacheOption="OnLoad" /></Image.Source></Image></InlineUIContainer><Run FontSize="26.6766666666667"> R is called open, if for each </Run><Run FontStyle="Italic" FontSize="26.6766666666667">x </Run><InlineUIContainer FontSize="14.6766666666667"><Image Stretch="Fill" Width="12.61" Height="10" InputMethod.IsInputMethodEnabled="True"><Image.Source><BitmapImage BaseUri="file:///C:/Users/wachs/AppData/Local/Temp/DyKnow/img30/2d1283a1-866f-4432-86dc-b343e7a26d38" UriSource="file:///C:/Users/wachs/AppData/Local/Temp/DyKnow/img30/2d1283a1-866f-4432-86dc-b343e7a26d38" CacheOption="OnLoad" /></Image.Source></Image></InlineUIContainer><Run FontStyle="Italic" FontSize="26.6766666666667"> U</Run><Run FontSize="26.6766666666667"> there exists and </Run><InlineUIContainer FontSize="14.6766666666667"><Image Stretch="Fill" Width="10" Height="10" InputMethod.IsInputMethodEnabled="True"><Image.Source><BitmapImage BaseUri="file:///C:/Users/wachs/AppData/Local/Temp/DyKnow/img30/b62b6b58-2ad5-4419-9118-7e41cac63498" UriSource="file:///C:/Users/wachs/AppData/Local/Temp/DyKnow/img30/b62b6b58-2ad5-4419-9118-7e41cac63498" CacheOption="OnLoad" /></Image.Source></Image></InlineUIContainer><Run FontStyle="Italic" FontSize="26.6766666666667"> &gt; 0</Run><Run FontSize="26.6766666666667"> such that the interval </Run><Run FontStyle="Italic" FontSize="26.6766666666667">( x - </Run><InlineUIContainer FontSize="14.6766666666667"><Image Stretch="Fill" Width="10" Height="10" InputMethod.IsInputMethodEnabled="True"><Image.Source><BitmapImage BaseUri="file:///C:/Users/wachs/AppData/Local/Temp/DyKnow/img30/b62b6b58-2ad5-4419-9118-7e41cac63498" UriSource="file:///C:/Users/wachs/AppData/Local/Temp/DyKnow/img30/b62b6b58-2ad5-4419-9118-7e41cac63498" CacheOption="OnLoad" /></Image.Source></Image></InlineUIContainer><Run FontStyle="Italic" FontSize="26.6766666666667">, x + </Run><InlineUIContainer FontSize="14.6766666666667"><Image Stretch="Fill" Width="10" Height="10" InputMethod.IsInputMethodEnabled="True"><Image.Source><BitmapImage BaseUri="file:///C:/Users/wachs/AppData/Local/Temp/DyKnow/img30/b62b6b58-2ad5-4419-9118-7e41cac63498" UriSource="file:///C:/Users/wachs/AppData/Local/Temp/DyKnow/img30/b62b6b58-2ad5-4419-9118-7e41cac63498" CacheOption="OnLoad" /></Image.Source></Image></InlineUIContainer><Run FontStyle="Italic" FontSize="26.6766666666667">)</Run><Run FontSize="26.6766666666667"> is contained in U. Such an interval is often called an </Run><InlineUIContainer FontSize="14.6766666666667"><Image Stretch="Fill" Width="10" Height="10" InputMethod.IsInputMethodEnabled="True"><Image.Source><BitmapImage BaseUri="file:///C:/Users/wachs/AppData/Local/Temp/DyKnow/img30/b62b6b58-2ad5-4419-9118-7e41cac63498" UriSource="file:///C:/Users/wachs/AppData/Local/Temp/DyKnow/img30/b62b6b58-2ad5-4419-9118-7e41cac63498" CacheOption="OnLoad" /></Image.Source></Image></InlineUIContainer><Run FontSize="26.6766666666667"> -neighborhood of </Run><Run FontStyle="Italic" FontSize="26.6766666666667">x</Run><Run FontSize="26.6766666666667">, or simply a neighborhood of </Run><Run FontStyle="Italic" FontSize="26.6766666666667">x</Run><Run FontSize="26.6766666666667">.</Run></Paragraph><Paragraph Margin="0,0,0,0" LineHeight="Auto" FontFamily="Trebuchet MS" FontWeight="Bold" FontSize="14.6666666666667" Background="#FFFFFFFF"><Run FontSize="26.6766666666667" xml:space="preserve" /></Paragraph><Paragraph Margin="0,6.67,0,6.67" LineHeight="Auto" FontFamily="Trebuchet MS" FontWeight="Bold" FontSize="14.6666666666667" Background="#FFFFFFFF"><Run FontSize="26.6766666666667">A set F is called closed if the complement of F, R \ F, is open.</Run></Paragraph></FlowDocument><FlowDocument FontFamily="Arial" FontStyle="Normal" FontWeight="Normal" FontSize="29.3433333333333" Foreground="#FF000000" PagePadding="5,0,5,0" AllowDrop="True" NumberSubstitution.CultureSource="User" xmlns="http://schemas.microsoft.com/winfx/2006/xaml/presentation" xmlns:x="http://schemas.microsoft.com/winfx/2006/xaml"><Paragraph Margin="0,0,0,0" LineHeight="Auto" FontFamily="Times New Roman" FontSize="16"><Run FontFamily="Trebuchet MS" FontWeight="Bold" FontSize="26.6766666666667">Example:</Run><Run FontFamily="Trebuchet MS" FontSize="26.6766666666667" xml:space="preserve"> Which of the following sets are open, closed, both, or neither ?</Run></Paragraph><List MarkerStyle="Decimal" MarkerOffset="17" Margin="0,0,0,0" Padding="0,0,0,0" FontFamily="Trebuchet MS" FontSize="26.6766666666667"><ListItem Margin="48,0,0,0" Padding="0,0,0,0" BorderThickness="1,1,1,1" FontSize="14.6766666666667"><Paragraph Margin="0,6.67,0,6.67" LineHeight="Auto" FontSize="14.6666666666667" Background="#FFEDF2F2"><Run FontSize="26.6766666666667">The intervals (-3, 3), [4, 7], (-4, 5], (0, </Run><InlineUIContainer FontSize="14.6766666666667"><Image Stretch="Fill" Width="12.61" Height="8.64" InputMethod.IsInputMethodEnabled="True"><Image.Source><BitmapImage BaseUri="file:///C:/Users/wachs/AppData/Local/Temp/DyKnow/img30/eac83b4f-4eae-4100-9c4c-7c2fa98ccbd6" UriSource="file:///C:/Users/wachs/AppData/Local/Temp/DyKnow/img30/eac83b4f-4eae-4100-9c4c-7c2fa98ccbd6" CacheOption="OnLoad" /></Image.Source></Image></InlineUIContainer><Run FontSize="26.6766666666667">) and [0, </Run><InlineUIContainer FontSize="14.6766666666667"><Image Stretch="Fill" Width="12.61" Height="8.64" InputMethod.IsInputMethodEnabled="True"><Image.Source><BitmapImage BaseUri="file:///C:/Users/wachs/AppData/Local/Temp/DyKnow/img30/eac83b4f-4eae-4100-9c4c-7c2fa98ccbd6" UriSource="file:///C:/Users/wachs/AppData/Local/Temp/DyKnow/img30/eac83b4f-4eae-4100-9c4c-7c2fa98ccbd6" CacheOption="OnLoad" /></Image.Source></Image></InlineUIContainer><Run FontSize="26.6766666666667">)</Run></Paragraph><Paragraph Margin="0,6.67,0,6.67" LineHeight="Auto" FontSize="14.6666666666667" Background="#FFEDF2F2"><Run FontSize="26.6766666666667" xml:space="preserve" /></Paragraph></ListItem><ListItem Margin="48,0,0,0" Padding="0,0,0,0" BorderThickness="1,1,1,1" FontSize="14.6766666666667"><Paragraph Margin="0,6.67,0,6.67" LineHeight="Auto" FontSize="14.6666666666667" Background="#FFEDF2F2"><Run FontSize="26.6766666666667">The sets R (the whole real line) and 0 (the empty set)</Run></Paragraph><Paragraph Margin="0,6.67,0,6.67" LineHeight="Auto" FontSize="14.6666666666667" Background="#FFEDF2F2"><Run FontSize="26.6766666666667" xml:space="preserve" /></Paragraph></ListItem><ListItem Margin="48,0,0,0" Padding="0,0,0,0" BorderThickness="1,1,1,1" FontSize="14.6766666666667"><Paragraph Margin="0,6.67,0,6.67" LineHeight="Auto" FontSize="14.6666666666667" Background="#FFEDF2F2"><Run FontSize="26.6766666666667">The set {1, 1/2, 1/3, 1/4, 1/5, ...} and {1, 1/2, 1/3, 1/4, ...} </Run><InlineUIContainer FontSize="14.6766666666667"><Image Stretch="Fill" Width="12" Height="10" InputMethod.IsInputMethodEnabled="True"><Image.Source><BitmapImage BaseUri="file:///C:/Users/wachs/AppData/Local/Temp/DyKnow/img30/d42fdf07-15ea-452c-ba06-0d503deeec90" UriSource="file:///C:/Users/wachs/AppData/Local/Temp/DyKnow/img30/d42fdf07-15ea-452c-ba06-0d503deeec90" CacheOption="OnLoad" /></Image.Source></Image></InlineUIContainer><Run FontFamily="Trebuchet MS" FontSize="26.6766666666667">{0}</Run></Paragraph></ListItem></List></FlowDocument>{\rtf1\ansi\ansicpg1252\uc1\htmautsp\deff2{\fonttbl{\f0\fcharset0 Times New Roman;}{\f2\fcharset0 Trebuchet MS;}}{\colortbl\red0\green0\blue0;\red255\green255\blue255;\red128\green0\blue128;} {\*\listtable {\list\listtemplateid1\listhybrid {\listlevel\levelnfc23\levelnfcn23\leveljc0\leveljcn0\levelfollow0\levelstartat1\levelspace0\levelindent0{\leveltext\leveltemplateid5\'01\'b7}{\levelnumbers;}\fi-360\li720\lin720\jclisttab\tx720} {\listlevel\levelnfc0\levelnfcn0\leveljc0\leveljcn0\levelfollow0\levelstartat1\levelspace0\levelindent0{\leveltext\leveltemplateid6\'02\'01.;}{\levelnumbers\'01;}\fi-360\li1440\lin1440\jclisttab\tx1440} {\listlevel\levelnfc0\levelnfcn0\leveljc0\leveljcn0\levelfollow0\levelstartat1\levelspace0\levelindent0{\leveltext\leveltemplateid7\'02\'02.;}{\levelnumbers\'01;}\fi-360\li2160\lin2160\jclisttab\tx2160} {\listlevel\levelnfc0\levelnfcn0\leveljc0\leveljcn0\levelfollow0\levelstartat1\levelspace0\levelindent0{\leveltext\leveltemplateid8\'02\'03.;}{\levelnumbers\'01;}\fi-360\li2880\lin2880\jclisttab\tx2880} {\listlevel\levelnfc0\levelnfcn0\leveljc0\leveljcn0\levelfollow0\levelstartat1\levelspace0\levelindent0{\leveltext\leveltemplateid9\'02\'04.;}{\levelnumbers\'01;}\fi-360\li3600\lin3600\jclisttab\tx3600} {\listlevel\levelnfc0\levelnfcn0\leveljc0\leveljcn0\levelfollow0\levelstartat1\levelspace0\levelindent0{\leveltext\leveltemplateid10\'02\'05.;}{\levelnumbers\'01;}\fi-360\li4320\lin4320\jclisttab\tx4320} {\listlevel\levelnfc0\levelnfcn0\leveljc0\leveljcn0\levelfollow0\levelstartat1\levelspace0\levelindent0{\leveltext\leveltemplateid11\'02\'06.;}{\levelnumbers\'01;}\fi-360\li5040\lin5040\jclisttab\tx5040} {\listlevel\levelnfc0\levelnfcn0\leveljc0\leveljcn0\levelfollow0\levelstartat1\levelspace0\levelindent0{\leveltext\leveltemplateid12\'02\'07.;}{\levelnumbers\'01;}\fi-360\li5760\lin5760\jclisttab\tx5760} {\listlevel\levelnfc0\levelnfcn0\leveljc0\leveljcn0\levelfollow0\levelstartat1\levelspace0\levelindent0{\leveltext\leveltemplateid13\'02\'08.;}{\levelnumbers\'01;}\fi-360\li6480\lin6480\jclisttab\tx6480} {\listname ;}\listid1}} {\*\listoverridetable {\listoverride\listid1\listoverridecount0\ls1} } \loch\hich\dbch\pard\tx1700\tx3400\tx5100\plain\ltrpar\itap0{\lang1033\fs40\f2\b\cf2 \cf2\ql{\f2 {\b0\cf0\ltrch It is fairly clear that when combining two open sets (either via union or intersection) the resulting set is again open, and the same statement should be true for closed sets. What about combining infinitely many sets ?}\li0\ri0\sa0\sb0\fi0\ql\par} {\f2 {\b0\cf0\ltrch }\li0\ri0\sa0\sb0\fi0\ql\par} {\f2 {\b0\cf0\ltrch }\li0\ri0\sa0\sb0\fi0\ql\par} {\f2 {\cf0\ltrch Proposition}{\b0\cf0\ltrch : Unions of Open Sets, Intersections of Closed Sets}\li0\ri0\sa0\sb0\fi0\ql\par} {\f2 {\b0\cf0\ltrch \tab }\li0\ri0\sa0\sb0\fi0\ql\par} {\cf2 {\pntext \'B7\tab}{\*\pn\pnlvlblt\pnstart1{\pntxtb\'B7}}{\b0\cf0\ltrch Every union of open sets is again open.}\li575\ri0\sa0\sb0\jclisttab\tx575\fi-360\ql\par} {\cf2 {\pntext \'B7\tab}{\*\pn\pnlvlblt\pnstart1{\pntxtb\'B7}}{\b0\cf0\ltrch Every intersection of closed sets is again closed.}\li575\ri0\sa0\sb0\jclisttab\tx575\fi-360\ql\par} {\cf2 {\pntext \'B7\tab}{\*\pn\pnlvlblt\pnstart1{\pntxtb\'B7}}{\b0\cf0\ltrch Every finite intersection of open sets is again open}\li575\ri0\sa0\sb0\jclisttab\tx575\fi-360\ql\par} {\cf2 {\pntext \'B7\tab}{\*\pn\pnlvlblt\pnstart1{\pntxtb\'B7}}{\b0\cf0\ltrch Every finite union of closed sets is again closed.}\li575\ri0\sa0\sb0\jclisttab\tx575\fi-360\ql\par} {\f2 {\b0\cf0\ltrch }\li0\ri0\sa0\sb0\fi0\ql\par} {\f2 {\b0\cf0\ltrch }\li0\ri0\sa0\sb0\fi0\ql\par} {\f2 {\cf0\ltrch Example:}\li0\ri0\sa0\sb0\fi0\ql\par} {\b\cf2 {\pntext \'B7\tab}{\*\pn\pnlvlblt\pnstart1{\pntxtb\'B7}}{\b0\cf0\ltrch Find a (countable) union of closed sets that is not closed}\li575\ri0\sa0\sb0\jclisttab\tx575\fi-360\ql\par} {\f2 {\b0\cf0\ltrch }\li0\ri0\sa0\sb0\fi0\ql\par} {\b\cf2 {\pntext \'B7\tab}{\*\pn\pnlvlblt\pnstart1{\pntxtb\'B7}}{\b0\cf0\ltrch Find a (countable) intersection of open sets that is not open}\li575\ri0\sa0\sb0\jclisttab\tx575\fi-360\ql\par} } }<FlowDocument FontFamily="Trebuchet MS" FontStyle="Normal" FontWeight="Bold" FontSize="26.6766666666667" Foreground="#FF800080" PagePadding="5,0,5,0" AllowDrop="True" NumberSubstitution.CultureSource="User" xmlns="http://schemas.microsoft.com/winfx/2006/xaml/presentation" xmlns:s="clr-namespace:System;assembly=mscorlib"><Paragraph><Run FontWeight="Normal" Foreground="#FF000000">It is fairly clear that when combining two open sets (either via union or intersection) the resulting set is again open, and the same statement should be true for closed sets. What about combining infinitely many sets ?</Run></Paragraph><Paragraph LineHeight="Auto"><Run FontWeight="Normal" Foreground="#FF000000" xml:space="preserve" /></Paragraph><Paragraph LineHeight="Auto"><Run FontWeight="Normal" Foreground="#FF000000" xml:space="preserve" /></Paragraph><Paragraph><Run Foreground="#FF000000">Proposition</Run><Run FontWeight="Normal" Foreground="#FF000000">: Unions of Open Sets, Intersections of Closed Sets</Run></Paragraph><Paragraph><Run FontWeight="Normal" Foreground="#FF000000" xml:space="preserve"> </Run></Paragraph><List MarkerStyle="Disc" MarkerOffset="17" Margin="0,0,0,0" Padding="0,0,0,0" FontFamily="Trebuchet MS" FontStyle="Normal" FontWeight="Normal" FontSize="26.6766666666667" Foreground="#FF000000"><List.Tag><s:Int32>0</s:Int32></List.Tag><ListItem Margin="38.33,0,0,0" Padding="0,0,0,0" BorderThickness="1,1,1,1" FontWeight="Bold"><Paragraph Margin="0,0,0,0" LineHeight="Auto" FontFamily="Trebuchet MS" FontStyle="Normal" FontWeight="Bold" FontSize="26.6766666666667" Foreground="#FF800080"><Run FontWeight="Normal" Foreground="#FF000000">Every union of open sets is again open.</Run></Paragraph></ListItem><ListItem Margin="38.33,0,0,0" Padding="0,0,0,0" BorderThickness="1,1,1,1" FontWeight="Bold"><Paragraph Margin="0,0,0,0" LineHeight="Auto" FontFamily="Trebuchet MS" FontStyle="Normal" FontWeight="Bold" FontSize="26.6766666666667" Foreground="#FF800080"><Run FontWeight="Normal" Foreground="#FF000000">Every intersection of closed sets is again closed.</Run></Paragraph></ListItem><ListItem Margin="38.33,0,0,0" Padding="0,0,0,0" BorderThickness="1,1,1,1" FontWeight="Bold"><Paragraph Margin="0,0,0,0" LineHeight="Auto" FontFamily="Trebuchet MS" FontStyle="Normal" FontWeight="Bold" FontSize="26.6766666666667" Foreground="#FF800080"><Run FontWeight="Normal" Foreground="#FF000000">Every finite intersection of open sets is again open</Run></Paragraph></ListItem><ListItem Margin="38.33,0,0,0" Padding="0,0,0,0" BorderThickness="1,1,1,1" FontWeight="Bold"><Paragraph Margin="0,0,0,0" LineHeight="Auto" FontFamily="Trebuchet MS" FontStyle="Normal" FontWeight="Bold" FontSize="26.6766666666667" Foreground="#FF800080"><Run FontWeight="Normal" Foreground="#FF000000">Every finite union of closed sets is again closed.</Run></Paragraph></ListItem></List><Paragraph><Run FontWeight="Normal" Foreground="#FF000000" xml:space="preserve" /></Paragraph><Paragraph LineHeight="Auto"><Run FontWeight="Normal" Foreground="#FF000000" xml:space="preserve" /></Paragraph><Paragraph LineHeight="Auto"><Run Foreground="#FF000000">Example:</Run></Paragraph><List MarkerStyle="Disc" MarkerOffset="17" Margin="0,0,0,0" Padding="0,0,0,0" FontFamily="Trebuchet MS" FontStyle="Normal" FontWeight="Bold" FontSize="26.6766666666667" Foreground="#FF000000"><List.Tag><s:Int32>0</s:Int32></List.Tag><ListItem Margin="38.33,0,0,0" Padding="0,0,0,0" BorderThickness="1,1,1,1" FontWeight="Normal"><Paragraph Margin="0,0,0,0" LineHeight="Auto" FontFamily="Trebuchet MS" FontStyle="Normal" FontWeight="Bold" FontSize="26.6766666666667" Foreground="#FF800080"><Run FontWeight="Normal" Foreground="#FF000000">Find a (countable) union of closed sets that is not closed</Run></Paragraph></ListItem></List><Paragraph LineHeight="Auto"><Run FontWeight="Normal" Foreground="#FF000000" xml:space="preserve" /></Paragraph><List MarkerStyle="Disc" MarkerOffset="17" Margin="0,0,0,0" Padding="0,0,0,0" FontFamily="Trebuchet MS" FontStyle="Normal" FontWeight="Normal" FontSize="26.6766666666667" Foreground="#FF000000"><List.Tag><s:Int32>0</s:Int32></List.Tag><ListItem Margin="38.33,0,0,0" Padding="0,0,0,0" BorderThickness="1,1,1,1"><Paragraph Margin="0,0,0,0" LineHeight="Auto" FontFamily="Trebuchet MS" FontStyle="Normal" FontWeight="Bold" FontSize="26.6766666666667" Foreground="#FF800080"><Run FontWeight="Normal" Foreground="#FF000000">Find a (countable) intersection of open sets that is not open</Run></Paragraph></ListItem></List></FlowDocument>2624719209<FlowDocument FontFamily="Trebuchet MS" FontStyle="Normal" FontWeight="Bold" FontSize="26.6766666666667" Foreground="#FF800080" PagePadding="5,0,5,0" AllowDrop="True" NumberSubstitution.CultureSource="User" xmlns="http://schemas.microsoft.com/winfx/2006/xaml/presentation" xmlns:x="http://schemas.microsoft.com/winfx/2006/xaml" xmlns:s="clr-namespace:System;assembly=mscorlib"><Paragraph><Run FontWeight="Normal" Foreground="#FF000000">It is fairly clear that when combining two open sets (either via union or intersection) the resulting set is again open, and the same statement should be true for closed sets. What about combining infinitely many sets ?</Run></Paragraph><Paragraph LineHeight="Auto"><Run FontWeight="Normal" Foreground="#FF000000" xml:space="preserve" /></Paragraph><Paragraph LineHeight="Auto"><Run FontWeight="Normal" Foreground="#FF000000" xml:space="preserve" /></Paragraph><Paragraph><Run Foreground="#FF000000">Proposition</Run><Run FontWeight="Normal" Foreground="#FF000000">: (Unions of Open Sets, Intersections of Closed Sets)</Run></Paragraph><Paragraph><Run FontWeight="Normal" Foreground="#FF000000" xml:space="preserve"> </Run></Paragraph><List MarkerStyle="Disc" MarkerOffset="17" Margin="0,0,0,0" Padding="0,0,0,0" FontFamily="Trebuchet MS" FontStyle="Normal" FontWeight="Normal" FontSize="26.6766666666667" Foreground="#FF000000"><List.Tag><s:Int32>0</s:Int32></List.Tag><ListItem Margin="38.33,0,0,0" Padding="0,0,0,0" BorderThickness="1,1,1,1" FontWeight="Bold"><Paragraph Margin="0,0,0,0" LineHeight="Auto" FontFamily="Trebuchet MS" FontStyle="Normal" FontWeight="Bold" FontSize="26.6766666666667" Foreground="#FF800080"><Run FontWeight="Normal" Foreground="#FF000000">Every union of open sets is again open.</Run></Paragraph></ListItem><ListItem Margin="38.33,0,0,0" Padding="0,0,0,0" BorderThickness="1,1,1,1" FontWeight="Bold"><Paragraph Margin="0,0,0,0" LineHeight="Auto" FontFamily="Trebuchet MS" FontStyle="Normal" FontWeight="Bold" FontSize="26.6766666666667" Foreground="#FF800080"><Run FontWeight="Normal" Foreground="#FF000000">Every intersection of closed sets is again closed.</Run></Paragraph></ListItem><ListItem Margin="38.33,0,0,0" Padding="0,0,0,0" BorderThickness="1,1,1,1" FontWeight="Bold"><Paragraph Margin="0,0,0,0" LineHeight="Auto" FontFamily="Trebuchet MS" FontStyle="Normal" FontWeight="Bold" FontSize="26.6766666666667" Foreground="#FF800080"><Run FontWeight="Normal" Foreground="#FF000000">Every finite intersection of open sets is again open</Run></Paragraph></ListItem><ListItem Margin="38.33,0,0,0" Padding="0,0,0,0" BorderThickness="1,1,1,1" FontWeight="Bold"><Paragraph Margin="0,0,0,0" LineHeight="Auto" FontFamily="Trebuchet MS" FontStyle="Normal" FontWeight="Bold" FontSize="26.6766666666667" Foreground="#FF800080"><Run FontWeight="Normal" Foreground="#FF000000">Every finite union of closed sets is again closed.</Run></Paragraph></ListItem></List><Paragraph><Run FontWeight="Normal" Foreground="#FF000000" xml:space="preserve" /></Paragraph><Paragraph LineHeight="Auto"><Run FontWeight="Normal" Foreground="#FF000000" xml:space="preserve" /></Paragraph><Paragraph LineHeight="Auto"><Run Foreground="#FF000000">Example:</Run></Paragraph><List MarkerStyle="Disc" MarkerOffset="17" Margin="0,0,0,0" Padding="0,0,0,0" FontFamily="Trebuchet MS" FontStyle="Normal" FontWeight="Bold" FontSize="26.6766666666667" Foreground="#FF000000"><List.Tag><s:Int32>0</s:Int32></List.Tag><ListItem Margin="38.33,0,0,0" Padding="0,0,0,0" BorderThickness="1,1,1,1" FontWeight="Normal"><Paragraph Margin="0,0,0,0" LineHeight="Auto" FontFamily="Trebuchet MS" FontStyle="Normal" FontWeight="Bold" FontSize="26.6766666666667" Foreground="#FF800080"><Run FontWeight="Normal" Foreground="#FF000000">Find a (countable) union of closed sets that is not closed</Run></Paragraph></ListItem></List><Paragraph LineHeight="Auto"><Run FontWeight="Normal" Foreground="#FF000000" xml:space="preserve" /></Paragraph><Paragraph LineHeight="Auto"><Run FontWeight="Normal" Foreground="#FF000000" xml:space="preserve" /></Paragraph><List MarkerStyle="Disc" MarkerOffset="17" Margin="0,0,0,0" Padding="0,0,0,0" FontFamily="Trebuchet MS" FontStyle="Normal" FontWeight="Normal" FontSize="26.6766666666667" Foreground="#FF000000"><List.Tag><s:Int32>0</s:Int32></List.Tag><ListItem Margin="38.33,0,0,0" Padding="0,0,0,0" BorderThickness="1,1,1,1"><Paragraph Margin="0,0,0,0" LineHeight="Auto" FontFamily="Trebuchet MS" FontStyle="Normal" FontWeight="Bold" FontSize="26.6766666666667" Foreground="#FF800080"><Run FontWeight="Normal" Foreground="#FF000000">Find a (countable) intersection of open sets that is not open</Run></Paragraph></ListItem></List></FlowDocument><FlowDocument FontFamily="Trebuchet MS" FontStyle="Normal" FontWeight="Bold" FontSize="26.6766666666667" Foreground="#FF800080" PagePadding="5,0,5,0" AllowDrop="True" NumberSubstitution.CultureSource="User" xmlns="http://schemas.microsoft.com/winfx/2006/xaml/presentation" xmlns:x="http://schemas.microsoft.com/winfx/2006/xaml" xmlns:s="clr-namespace:System;assembly=mscorlib"><Paragraph><Run FontWeight="Normal" Foreground="#FF000000">It is fairly clear that when combining two open sets (either via union or intersection) the resulting set is again open, and the same statement should be true for closed sets. What about combining infinitely many sets ?</Run></Paragraph><Paragraph LineHeight="Auto"><Run FontWeight="Normal" Foreground="#FF000000" xml:space="preserve" /></Paragraph><Paragraph LineHeight="Auto"><Run FontWeight="Normal" Foreground="#FF000000" xml:space="preserve" /></Paragraph><Paragraph><Run Foreground="#FF000000">Proposition</Run><Run FontWeight="Normal" Foreground="#FF000000">: (Unions of Open Sets, Intersections of Closed Sets)</Run></Paragraph><Paragraph><Run FontWeight="Normal" Foreground="#FF000000" xml:space="preserve"> </Run></Paragraph><List MarkerStyle="Disc" MarkerOffset="17" Margin="0,0,0,0" Padding="0,0,0,0" FontFamily="Trebuchet MS" FontStyle="Normal" FontWeight="Normal" FontSize="26.6766666666667" Foreground="#FF000000"><List.Tag><s:Int32>0</s:Int32></List.Tag><ListItem Margin="38.33,0,0,0" Padding="0,0,0,0" BorderThickness="1,1,1,1" FontWeight="Bold"><Paragraph Margin="0,0,0,0" LineHeight="Auto" FontFamily="Trebuchet MS" FontStyle="Normal" FontWeight="Bold" FontSize="26.6766666666667" Foreground="#FF800080"><Run FontWeight="Normal" Foreground="#FF000000">Every union of open sets is again open.</Run></Paragraph></ListItem><ListItem Margin="38.33,0,0,0" Padding="0,0,0,0" BorderThickness="1,1,1,1" FontWeight="Bold"><Paragraph Margin="0,0,0,0" LineHeight="Auto" FontFamily="Trebuchet MS" FontStyle="Normal" FontWeight="Bold" FontSize="26.6766666666667" Foreground="#FF800080"><Run FontWeight="Normal" Foreground="#FF000000">Every intersection of closed sets is again closed.</Run></Paragraph></ListItem><ListItem Margin="38.33,0,0,0" Padding="0,0,0,0" BorderThickness="1,1,1,1" FontWeight="Bold"><Paragraph Margin="0,0,0,0" LineHeight="Auto" FontFamily="Trebuchet MS" FontStyle="Normal" FontWeight="Bold" FontSize="26.6766666666667" Foreground="#FF800080"><Run FontWeight="Normal" Foreground="#FF000000">Every finite intersection of open sets is again open</Run></Paragraph></ListItem><ListItem Margin="38.33,0,0,0" Padding="0,0,0,0" BorderThickness="1,1,1,1" FontWeight="Bold"><Paragraph Margin="0,0,0,0" LineHeight="Auto" FontFamily="Trebuchet MS" FontStyle="Normal" FontWeight="Bold" FontSize="26.6766666666667" Foreground="#FF800080"><Run FontWeight="Normal" Foreground="#FF000000">Every finite union of closed sets is again closed.</Run></Paragraph></ListItem></List><Paragraph><Run FontWeight="Normal" Foreground="#FF000000" xml:space="preserve" /></Paragraph><Paragraph LineHeight="Auto"><Run FontWeight="Normal" Foreground="#FF000000" xml:space="preserve" /></Paragraph><Paragraph LineHeight="Auto"><Run Foreground="#FF000000">Example:</Run></Paragraph><List MarkerStyle="Disc" MarkerOffset="17" Margin="0,0,0,0" Padding="0,0,0,0" FontFamily="Trebuchet MS" FontStyle="Normal" FontWeight="Bold" FontSize="26.6766666666667" Foreground="#FF000000"><List.Tag><s:Int32>0</s:Int32></List.Tag><ListItem Margin="38.33,0,0,0" Padding="0,0,0,0" BorderThickness="1,1,1,1" FontWeight="Normal"><Paragraph Margin="0,0,0,0" LineHeight="Auto" FontFamily="Trebuchet MS" FontStyle="Normal" FontWeight="Bold" FontSize="26.6766666666667" Foreground="#FF800080"><Run FontWeight="Normal" Foreground="#FF000000">Find a (countable) union of closed sets that is not closed</Run></Paragraph></ListItem></List><Paragraph LineHeight="Auto"><Run FontWeight="Normal" Foreground="#FF000000" xml:space="preserve" /></Paragraph><Paragraph LineHeight="Auto"><Run FontWeight="Normal" Foreground="#FF000000" xml:space="preserve" /></Paragraph><List MarkerStyle="Disc" MarkerOffset="17" Margin="0,0,0,0" Padding="0,0,0,0" FontFamily="Trebuchet MS" FontStyle="Normal" FontWeight="Normal" FontSize="26.6766666666667" Foreground="#FF000000"><List.Tag><s:Int32>0</s:Int32></List.Tag><ListItem Margin="38.33,0,0,0" Padding="0,0,0,0" BorderThickness="1,1,1,1"><Paragraph Margin="0,0,0,0" LineHeight="Auto" FontFamily="Trebuchet MS" FontStyle="Normal" FontWeight="Bold" FontSize="26.6766666666667" Foreground="#FF800080"><Run FontWeight="Normal" Foreground="#FF000000">Find a (countable) intersection of open sets that is not open</Run></Paragraph></ListItem></List><Paragraph Margin="0,0,0,0" LineHeight="Auto" FontFamily="Trebuchet MS" FontStyle="Normal" FontWeight="Bold" FontSize="26.6766666666667" Foreground="#FF800080"><Run FontWeight="Normal" Foreground="#FF000000" xml:space="preserve" /></Paragraph><Paragraph Margin="0,0,0,0" LineHeight="Auto" FontFamily="Trebuchet MS" FontStyle="Normal" FontWeight="Bold" FontSize="26.6766666666667" Foreground="#FF800080"><Run FontWeight="Normal" Foreground="#FF000000" xml:space="preserve" /></Paragraph><List MarkerStyle="Disc" MarkerOffset="17" Margin="0,0,0,0" Padding="0,0,0,0" FontFamily="Trebuchet MS" FontStyle="Normal" FontWeight="Normal" FontSize="26.6766666666667" Foreground="#FF000000"><List.Tag><s:Int32>0</s:Int32></List.Tag><ListItem Margin="38.33,0,0,0" Padding="0,0,0,0" BorderThickness="1,1,1,1"><Paragraph Margin="0,0,0,0" LineHeight="Auto" FontFamily="Trebuchet MS" FontStyle="Normal" FontWeight="Bold" FontSize="26.6766666666667" Foreground="#FF800080"><Run FontWeight="Normal" Foreground="#FF000000">Consider the collection of sets (0, 1/j) for all j &gt; 0. What is the intersection of all of these sets ?</Run></Paragraph></ListItem></List></FlowDocument>{\rtf1\ansi\ansicpg1252\uc1\htmautsp\deff2{\fonttbl{\f0\fcharset0 Times New Roman;}{\f2\fcharset0 Trebuchet MS;}}{\colortbl\red0\green0\blue0;\red255\green255\blue255;\red128\green0\blue128;}\loch\hich\dbch\pard\tx1700\tx3400\tx5100\plain\ltrpar\itap0{\lang1033\fs40\f2\b\cf2 \cf2\ql{\fs36\f2 {\cf0\ltrch Proof:}\li0\ri0\sa0\sb0\fi0\ql\par} {\fs36\f2 {\b0\cf0\ltrch }\li0\ri0\sa0\sb0\fi0\ql\par} {\fs36\f2 {\b0\cf0\ltrch Let \{ U n \} be a collection of open sets, and let U = U n. Take any x in U. Being in the union of all U's, it must be contained in one specific U n. Since that set is open, there exists a neighborhood of x contained in that specific U n. But then that neighborhood must also be contained in the union U. Hence, any x in U has a neighborhood that is also in U, which means by definition that U is open.}\li0\ri0\sa0\sb0\fi0\ql\par} {\fs36\f2 {\b0\cf0\ltrch }\li0\ri0\sa0\sb0\fi0\ql\par} {\fs36\f2 {\b0\cf0\ltrch To prove the second statement, simply use the de Morgan's laws.}\li0\ri0\sa0\sb0\fi0\ql\par} {\fs36\f2 {\b0\cf0\ltrch }\li0\ri0\sa0\sb0\fi0\ql\par} {\fs36\f2 {\b0\cf0\ltrch Now let U n, n=1, 2, 3, ..., N be finitely many open sets. Take x in the intersection of all of them. Then:}\li0\ri0\sa0\sb0\fi0\ql\par} {\fs36\f2 {\b0\cf0\ltrch }\li0\ri0\sa0\sb0\fi0\ql\par} {\fs36\f2 {\b0\cf0\ltrch x is in the first set: there exists an with ( x - , x + ) contained in the first set}\li0\ri0\sa0\sb0\fi0\ql\par} {\fs36\f2 {\b0\cf0\ltrch x is in the second set: there is with ( x - , x + ) contained in the second set.}\li0\ri0\sa0\sb0\fi0\ql\par} {\fs36\f2 {\b0\cf0\ltrch ....}\li0\ri0\sa0\sb0\fi0\ql\par} {\fs36\f2 {\b0\cf0\ltrch x is in the N-th set: there is with ( x - , x + ) contained in the last set.}\li0\ri0\sa0\sb0\fi0\ql\par} {\fs36\f2 {\b0\cf0\ltrch But then}\li0\ri0\sa0\sb0\fi0\ql\par} {\fs36\f2 {\b0\cf0\ltrch let = min\{ , , ..., \}. Then ( x - , x + ) is contained in each set U n}\li0\ri0\sa0\sb0\fi0\ql\par} {\fs36\f2 {\b0\cf0\ltrch Hence it is contained in their finite intersection, which is therefore open, since x was arbitrary.}\li0\ri0\sa0\sb0\fi0\ql\par} } }<FlowDocument FontFamily="Trebuchet MS" FontStyle="Normal" FontWeight="Bold" FontSize="26.6766666666667" Foreground="#FF800080" PagePadding="5,0,5,0" AllowDrop="True" NumberSubstitution.CultureSource="User" xmlns="http://schemas.microsoft.com/winfx/2006/xaml/presentation"><Paragraph LineHeight="Auto"><Run FontSize="24.01" Foreground="#FF000000">Proof:</Run></Paragraph><Paragraph LineHeight="Auto"><Run FontWeight="Normal" FontSize="24.01" Foreground="#FF000000" xml:space="preserve" /></Paragraph><Paragraph LineHeight="Auto"><Run FontWeight="Normal" FontSize="24.01" Foreground="#FF000000" xml:space="preserve">Let { U n } be a collection of open sets, and let U = U n. Take any x in U. Being in the union of all U's, it must be contained in one specific U n. Since that set is open, there exists a neighborhood of x contained in that specific U n. But then that neighborhood must also be contained in the union U. Hence, any x in U has a neighborhood that is also in U, which means by definition that U is open.</Run></Paragraph><Paragraph LineHeight="Auto"><Run FontWeight="Normal" FontSize="24.01" Foreground="#FF000000" xml:space="preserve" /></Paragraph><Paragraph LineHeight="Auto"><Run FontWeight="Normal" FontSize="24.01" Foreground="#FF000000">To prove the second statement, simply use the de Morgan's laws.</Run></Paragraph><Paragraph LineHeight="Auto"><Run FontWeight="Normal" FontSize="24.01" Foreground="#FF000000" xml:space="preserve" /></Paragraph><Paragraph LineHeight="Auto"><Run FontWeight="Normal" FontSize="24.01" Foreground="#FF000000">Now let U n, n=1, 2, 3, ..., N be finitely many open sets. Take x in the intersection of all of them. Then:</Run></Paragraph><Paragraph LineHeight="Auto"><Run FontWeight="Normal" FontSize="24.01" Foreground="#FF000000" xml:space="preserve" /></Paragraph><Paragraph LineHeight="Auto"><Run FontWeight="Normal" FontSize="24.01" Foreground="#FF000000" xml:space="preserve">x is in the first set: there exists an with ( x - , x + ) contained in the first set</Run></Paragraph><Paragraph LineHeight="Auto"><Run FontWeight="Normal" FontSize="24.01" Foreground="#FF000000" xml:space="preserve">x is in the second set: there is with ( x - , x + ) contained in the second set.</Run></Paragraph><Paragraph LineHeight="Auto"><Run FontWeight="Normal" FontSize="24.01" Foreground="#FF000000">....</Run></Paragraph><Paragraph LineHeight="Auto"><Run FontWeight="Normal" FontSize="24.01" Foreground="#FF000000" xml:space="preserve">x is in the N-th set: there is with ( x - , x + ) contained in the last set.</Run></Paragraph><Paragraph LineHeight="Auto"><Run FontWeight="Normal" FontSize="24.01" Foreground="#FF000000">But then</Run></Paragraph><Paragraph LineHeight="Auto"><Run FontWeight="Normal" FontSize="24.01" Foreground="#FF000000" xml:space="preserve">let = min{ , , ..., }. Then ( x - , x + ) is contained in each set U n</Run></Paragraph><Paragraph LineHeight="Auto"><Run FontWeight="Normal" FontSize="24.01" Foreground="#FF000000">Hence it is contained in their finite intersection, which is therefore open, since x was arbitrary.</Run></Paragraph></FlowDocument>2624719209<FlowDocument FontFamily="Trebuchet MS" FontStyle="Normal" FontWeight="Bold" FontSize="26.6766666666667" Foreground="#FF800080" PagePadding="5,0,5,0" AllowDrop="True" NumberSubstitution.CultureSource="User" xmlns="http://schemas.microsoft.com/winfx/2006/xaml/presentation" xmlns:x="http://schemas.microsoft.com/winfx/2006/xaml"><Paragraph LineHeight="Auto"><Run FontSize="24.01" Foreground="#FF000000">Proof:</Run></Paragraph><Paragraph LineHeight="Auto"><Run FontWeight="Normal" FontSize="24.01" Foreground="#FF000000" xml:space="preserve" /></Paragraph><Paragraph LineHeight="Auto"><Run FontWeight="Normal" FontSize="24.01" Foreground="#FF000000" xml:space="preserve">Let { U n } be a collection of open sets, and let U = U n. Take any x in U. Being in the union of all U's, it must be contained in one specific U n. Since that set is open, there exists a neighborhood of x contained in that specific U n. But then that neighborhood must also be contained in the union U. Hence, any x in U has a neighborhood that is also in U, which means by definition that U is open.</Run></Paragraph><Paragraph LineHeight="Auto"><Run FontWeight="Normal" FontSize="24.01" Foreground="#FF000000" xml:space="preserve" /></Paragraph><Paragraph LineHeight="Auto"><Run FontWeight="Normal" FontSize="24.01" Foreground="#FF000000">To prove the second statement, simply use the de Morgan's laws.</Run></Paragraph><Paragraph LineHeight="Auto"><Run FontWeight="Normal" FontSize="24.01" Foreground="#FF000000" xml:space="preserve" /></Paragraph><Paragraph LineHeight="Auto"><Run FontWeight="Normal" FontSize="24.01" Foreground="#FF000000">Now let U n, n=1, 2, 3, ..., N be finitely many open sets. Take x in the intersection of all of them. Then:</Run></Paragraph><Paragraph LineHeight="Auto"><Run FontWeight="Normal" FontSize="24.01" Foreground="#FF000000" xml:space="preserve" /></Paragraph><Paragraph LineHeight="Auto"><Run FontWeight="Normal" FontSize="24.01" Foreground="#FF000000" xml:space="preserve">x is in the first set: there exists an with ( x - , x + ) contained in the first set</Run></Paragraph><Paragraph LineHeight="Auto"><Run FontWeight="Normal" FontSize="24.01" Foreground="#FF000000" xml:space="preserve">x is in the second set: there is with ( x - , x + ) contained in the second set.</Run></Paragraph><Paragraph LineHeight="Auto"><Run FontWeight="Normal" FontSize="24.01" Foreground="#FF000000">....</Run></Paragraph><Paragraph LineHeight="Auto"><Run FontWeight="Normal" FontSize="24.01" Foreground="#FF000000" xml:space="preserve">x is in the N-th set: there is with ( x - , x + ) contained in the last set.</Run></Paragraph><Paragraph LineHeight="Auto"><Run FontWeight="Normal" FontSize="24.01" Foreground="#FF000000" xml:space="preserve" /></Paragraph><Paragraph LineHeight="Auto"><Run FontWeight="Normal" FontSize="24.01" Foreground="#FF000000" xml:space="preserve">But then let = min{ , , ..., }. Then ( x - , x + ) is contained in each set U n</Run></Paragraph><Paragraph LineHeight="Auto"><Run FontWeight="Normal" FontSize="24.01" Foreground="#FF000000" xml:space="preserve" /></Paragraph><Paragraph LineHeight="Auto"><Run FontWeight="Normal" FontSize="24.01" Foreground="#FF000000">Hence it is contained in their finite intersection, which is therefore open, since x was arbitrary.</Run></Paragraph></FlowDocument>{\rtf1\ansi\ansicpg1252\uc1\htmautsp\deff2{\fonttbl{\f0\fcharset0 Times New Roman;}{\f2\fcharset0 Trebuchet MS;}}{\colortbl\red0\green0\blue0;\red255\green255\blue255;}\loch\hich\dbch\pard\tx1700\tx3400\tx5100\plain\ltrpar\itap0{\lang1033\fs36\f2\b\cf0 \cf0\ql{\fs40\f2 {\b0\ltrch How complicated can an open or closed set really be ? The basic open (or closed) sets in the real line are the intervals, and they are certainly not complicated. As it will turn out, open sets in the real line are generally easy, while closed sets can be very complicated.}\li0\ri0\sa0\sb0\fi0\ql\par} {\fs40\f2 {\b0\ltrch }\li0\ri0\sa0\sb0\fi0\ql\par} {\fs40\f2 {\b0\ltrch The worst-case scenario for the open sets, in fact, will be given in the next result, and we will concentrate on closed sets for much of the rest of this chapter.}\li0\ri0\sa0\sb0\fi0\ql\par} {\fs40\f2 {\b0\ltrch }\li0\ri0\sa0\sb0\fi0\ql\par} {\fs40\f2 {\ltrch Proposition: }{\b0\ltrch (Characterizing Open Sets)}\li0\ri0\sa0\sb0\fi0\ql\par} {\fs40\f2 {\b0\ltrch \tab Let U R be an arbitrary open set. Then there are countably many pairwise disjoint open intervals Un such that U = Un}\li0\ri0\sa0\sb0\fi0\ql\par} {\fs40\f2 {\b0\ltrch }\li0\ri0\sa0\sb0\fi0\ql\par} } }<FlowDocument FontFamily="Trebuchet MS" FontStyle="Normal" FontWeight="Bold" FontSize="24.01" Foreground="#FF000000" PagePadding="5,0,5,0" AllowDrop="True" NumberSubstitution.CultureSource="User" xmlns="http://schemas.microsoft.com/winfx/2006/xaml/presentation"><Paragraph LineHeight="Auto"><Run FontWeight="Normal" FontSize="26.6766666666667">How complicated can an open or closed set really be ? The basic open (or closed) sets in the real line are the intervals, and they are certainly not complicated. As it will turn out, open sets in the real line are generally easy, while closed sets can be very complicated.</Run></Paragraph><Paragraph LineHeight="Auto"><Run FontWeight="Normal" FontSize="26.6766666666667" xml:space="preserve" /></Paragraph><Paragraph LineHeight="Auto"><Run FontWeight="Normal" FontSize="26.6766666666667">The worst-case scenario for the open sets, in fact, will be given in the next result, and we will concentrate on closed sets for much of the rest of this chapter.</Run></Paragraph><Paragraph LineHeight="Auto"><Run FontWeight="Normal" FontSize="26.6766666666667" xml:space="preserve" /></Paragraph><Paragraph LineHeight="Auto"><Run FontSize="26.6766666666667" xml:space="preserve">Proposition: </Run><Run FontWeight="Normal" FontSize="26.6766666666667" xml:space="preserve"> (Characterizing Open Sets)</Run></Paragraph><Paragraph LineHeight="Auto"><Run FontWeight="Normal" FontSize="26.6766666666667" xml:space="preserve"> Let U R be an arbitrary open set. Then there are countably many pairwise disjoint open intervals Un such that U = Un</Run></Paragraph><Paragraph LineHeight="Auto"><Run FontWeight="Normal" FontSize="26.6766666666667" xml:space="preserve" /></Paragraph></FlowDocument>2624719209{\rtf1\ansi\ansicpg1252\uc1\htmautsp\deff2{\fonttbl{\f0\fcharset0 Times New Roman;}{\f2\fcharset0 Trebuchet MS;}}{\colortbl\red0\green0\blue0;\red255\green255\blue255;}\loch\hich\dbch\pard\tx1700\tx3400\tx5100\plain\ltrpar\itap0{\lang1033\fs40\f2\b\cf0 \cf0\ql{\fs36\f2 {\ltrch Proof:}\li0\ri0\sa0\sb0\fi0\ql\par} {\fs36\f2 {\b0\ltrch }\li0\ri0\sa0\sb0\fi0\ql\par} {\fs36\f2 {\b0\ltrch This proposition is rather interesting, giving a complete description of any possible open set in the real line. To prove it, we will make use of equivalence relations and classes again. First, let us define a relation on U:}\li0\ri0\sa0\sb0\fi0\ql\par} {\fs36\f2 {\b0\ltrch }\li0\ri0\sa0\sb0\fi0\ql\par} {\fs36\f2 {\b0\ltrch if a and b are in U, we say that a ~ b if the whole line segment between a and b is also contained in U.}\li0\ri0\sa0\sb0\fi750\ql\par} {\fs36\f2 {\b0\ltrch }\li0\ri0\sa0\sb0\fi0\ql\par} {\fs36\f2 {\b0\ltrch This relation is an equivalence relation, so that we know immediately that U equals the union of the equivalence classes, and the equivalence classes are pairwise disjoint. Denote those equivalent classes by U n}\li0\ri0\sa0\sb0\fi0\ql\par} {\fs36\f2 {\b0\ltrch }\li0\ri0\sa0\sb0\fi0\ql\par} {\fs36\f2 {\b0\ltrch Each U n is an interval: take any two points a and b in U n. Being in the same equivalence classes, a and b must be related. But then the whole line segment between a and b is contained in U n as well. Since a and b were arbitrary, U n is indeed an interval.}\li0\ri0\sa0\sb0\fi0\ql\par} {\fs36\f2 {\b0\ltrch }\li0\ri0\sa0\sb0\fi0\ql\par} {\fs36\f2 {\b0\ltrch Each U n is open: take any x U n. Then x U, and since U is open, there exists an > 0 such that ( x - , x + ) is contained in U. But clearly each point in that interval is related to x, hence this neighborhood is contained in U n, proving that U n is open.}\li0\ri0\sa0\sb0\fi0\ql\par} {\fs36\f2 {\b0\ltrch }\li0\ri0\sa0\sb0\fi0\ql\par} {\fs36\f2 {\b0\ltrch There are only countably many U n: This seems the hard part. But, each U n must contain at least one different rational number. Since there are only countably many rational numbers, there can only be countably many of the U n's (since they are disjoint).}\li0\ri0\sa0\sb0\fi0\ql\par} } }<FlowDocument FontFamily="Trebuchet MS" FontStyle="Normal" FontWeight="Bold" FontSize="26.6766666666667" Foreground="#FF000000" PagePadding="5,0,5,0" AllowDrop="True" NumberSubstitution.CultureSource="User" xmlns="http://schemas.microsoft.com/winfx/2006/xaml/presentation"><Paragraph LineHeight="Auto"><Run FontSize="24.01">Proof:</Run></Paragraph><Paragraph LineHeight="Auto"><Run FontWeight="Normal" FontSize="24.01" xml:space="preserve" /></Paragraph><Paragraph LineHeight="Auto"><Run FontWeight="Normal" FontSize="24.01">This proposition is rather interesting, giving a complete description of any possible open set in the real line. To prove it, we will make use of equivalence relations and classes again. First, let us define a relation on U:</Run></Paragraph><Paragraph LineHeight="Auto"><Run FontWeight="Normal" FontSize="24.01" xml:space="preserve" /></Paragraph><Paragraph TextIndent="50" LineHeight="Auto"><Run FontWeight="Normal" FontSize="24.01">if a and b are in U, we say that a ~ b if the whole line segment between a and b is also contained in U.</Run></Paragraph><Paragraph LineHeight="Auto"><Run FontWeight="Normal" FontSize="24.01" xml:space="preserve" /></Paragraph><Paragraph LineHeight="Auto"><Run FontWeight="Normal" FontSize="24.01">This relation is an equivalence relation, so that we know immediately that U equals the union of the equivalence classes, and the equivalence classes are pairwise disjoint. Denote those equivalent classes by U n</Run></Paragraph><Paragraph LineHeight="Auto"><Run FontWeight="Normal" FontSize="24.01" xml:space="preserve" /></Paragraph><Paragraph LineHeight="Auto"><Run FontWeight="Normal" FontSize="24.01">Each U n is an interval: take any two points a and b in U n. Being in the same equivalence classes, a and b must be related. But then the whole line segment between a and b is contained in U n as well. Since a and b were arbitrary, U n is indeed an interval.</Run></Paragraph><Paragraph LineHeight="Auto"><Run FontWeight="Normal" FontSize="24.01" xml:space="preserve" /></Paragraph><Paragraph LineHeight="Auto"><Run FontWeight="Normal" FontSize="24.01" xml:space="preserve">Each U n is open: take any x U n. Then x U, and since U is open, there exists an &gt; 0 such that ( x - , x + ) is contained in U. But clearly each point in that interval is related to x, hence this neighborhood is contained in U n, proving that U n is open.</Run></Paragraph><Paragraph LineHeight="Auto"><Run FontWeight="Normal" FontSize="24.01" xml:space="preserve" /></Paragraph><Paragraph LineHeight="Auto"><Run FontWeight="Normal" FontSize="24.01">There are only countably many U n: This seems the hard part. But, each U n must contain at least one different rational number. Since there are only countably many rational numbers, there can only be countably many of the U n's (since they are disjoint).</Run></Paragraph></FlowDocument>2624719209{\rtf1\ansi\ansicpg1252\uc1\htmautsp\deff2{\fonttbl{\f0\fcharset0 Times New Roman;}{\f2\fcharset0 Trebuchet MS;}}{\colortbl\red0\green0\blue0;\red255\green255\blue255;} {\*\listtable {\list\listtemplateid1\listhybrid {\listlevel\levelnfc23\levelnfcn23\leveljc0\leveljcn0\levelfollow0\levelstartat1\levelspace0\levelindent0{\leveltext\leveltemplateid5\'01\'b7}{\levelnumbers;}\fi-360\li720\lin720\jclisttab\tx720} {\listlevel\levelnfc0\levelnfcn0\leveljc0\leveljcn0\levelfollow0\levelstartat1\levelspace0\levelindent0{\leveltext\leveltemplateid6\'02\'01.;}{\levelnumbers\'01;}\fi-360\li1440\lin1440\jclisttab\tx1440} {\listlevel\levelnfc0\levelnfcn0\leveljc0\leveljcn0\levelfollow0\levelstartat1\levelspace0\levelindent0{\leveltext\leveltemplateid7\'02\'02.;}{\levelnumbers\'01;}\fi-360\li2160\lin2160\jclisttab\tx2160} {\listlevel\levelnfc0\levelnfcn0\leveljc0\leveljcn0\levelfollow0\levelstartat1\levelspace0\levelindent0{\leveltext\leveltemplateid8\'02\'03.;}{\levelnumbers\'01;}\fi-360\li2880\lin2880\jclisttab\tx2880} {\listlevel\levelnfc0\levelnfcn0\leveljc0\leveljcn0\levelfollow0\levelstartat1\levelspace0\levelindent0{\leveltext\leveltemplateid9\'02\'04.;}{\levelnumbers\'01;}\fi-360\li3600\lin3600\jclisttab\tx3600} {\listlevel\levelnfc0\levelnfcn0\leveljc0\leveljcn0\levelfollow0\levelstartat1\levelspace0\levelindent0{\leveltext\leveltemplateid10\'02\'05.;}{\levelnumbers\'01;}\fi-360\li4320\lin4320\jclisttab\tx4320} {\listlevel\levelnfc0\levelnfcn0\leveljc0\leveljcn0\levelfollow0\levelstartat1\levelspace0\levelindent0{\leveltext\leveltemplateid11\'02\'06.;}{\levelnumbers\'01;}\fi-360\li5040\lin5040\jclisttab\tx5040} {\listlevel\levelnfc0\levelnfcn0\leveljc0\leveljcn0\levelfollow0\levelstartat1\levelspace0\levelindent0{\leveltext\leveltemplateid12\'02\'07.;}{\levelnumbers\'01;}\fi-360\li5760\lin5760\jclisttab\tx5760} {\listlevel\levelnfc0\levelnfcn0\leveljc0\leveljcn0\levelfollow0\levelstartat1\levelspace0\levelindent0{\leveltext\leveltemplateid13\'02\'08.;}{\levelnumbers\'01;}\fi-360\li6480\lin6480\jclisttab\tx6480} {\listname ;}\listid1}} {\*\listoverridetable {\listoverride\listid1\listoverridecount0\ls1} } \loch\hich\dbch\pard\tx1700\tx3400\tx5100\plain\ltrpar\itap0{\lang1033\fs36\f2\cf0 \cf0\ql{\f2 {\b\ltrch Definition: (Boundary, Accumulation, Interior, and Isolated Points)}\li0\ri0\sa0\sb0\fi0\ql\par} {\f2 {\ltrch Let S be an arbitrary set in the real line R.}\li0\ri0\sa0\sb0\fi0\ql\par} {\f2 {\ltrch }\li0\ri0\sa0\sb0\fi0\ql\par} {{\pntext \'B7\tab}{\*\pn\pnlvlblt\pnstart1{\pntxtb\'B7}}{\ltrch A point b in R is called boundary point of S if every non-empty neighborhood of b intersects S and the complement of S. The set of all boundary points of S is called the boundary of S, denoted by bd(S).}\li575\ri0\sa0\sb0\jclisttab\tx575\fi-360\ql\par} {{\ltrch }\li575\ri0\sa0\sb0\fi0\ql\par} {{\pntext \'B7\tab}{\*\pn\pnlvlblt\pnstart1{\pntxtb\'B7}}{\ltrch A point s in S is called interior point of S if there exists a neighborhood of S completely contained in S. The set of all interior points of S is called the interior, denoted by }\line {\ltrch int(S).}\li575\ri0\sa0\sb0\jclisttab\tx575\fi-360\ql\par} {{\ltrch }\li575\ri0\sa0\sb0\fi0\ql\par} {{\pntext \'B7\tab}{\*\pn\pnlvlblt\pnstart1{\pntxtb\'B7}}{\ltrch A point t in S is called isolated point of S if there exists a neighborhood U of t such that U intersect S = \{t\}.}\li575\ri0\sa0\sb0\jclisttab\tx575\fi-360\ql\par} {{\ltrch }\li575\ri0\sa0\sb0\fi0\ql\par} {{\pntext \'B7\tab}{\*\pn\pnlvlblt\pnstart1{\pntxtb\'B7}}{\ltrch A point r in S is called accumulation point, if every neighborhood of r contains infinitely many distinct points of S.}\li575\ri0\sa0\sb0\jclisttab\tx575\fi-360\ql\par} } }<FlowDocument FontFamily="Trebuchet MS" FontStyle="Normal" FontWeight="Normal" FontSize="24.01" Foreground="#FF000000" PagePadding="5,0,5,0" AllowDrop="True" NumberSubstitution.CultureSource="User" xmlns="http://schemas.microsoft.com/winfx/2006/xaml/presentation" xmlns:s="clr-namespace:System;assembly=mscorlib"><Paragraph LineHeight="Auto"><Run FontWeight="Bold">Definition: (Boundary, Accumulation, Interior, and Isolated Points)</Run></Paragraph><Paragraph LineHeight="Auto">Let S be an arbitrary set in the real line R.</Paragraph><Paragraph LineHeight="Auto" xml:space="preserve" /><List MarkerStyle="Disc" MarkerOffset="17" Margin="0,0,0,0" Padding="0,0,0,0" FontFamily="Trebuchet MS" FontStyle="Normal" FontWeight="Normal" FontSize="24.01" Foreground="#FF000000"><List.Tag><s:Int32>0</s:Int32></List.Tag><ListItem Margin="38.33,0,0,0" Padding="0,0,0,0" BorderThickness="1,1,1,1"><Paragraph Margin="0,0,0,0" LineHeight="Auto" FontFamily="Trebuchet MS" FontStyle="Normal" FontWeight="Normal" FontSize="24.01" Foreground="#FF000000">A point b in R is called boundary point of S if every non-empty neighborhood of b intersects S and the complement of S. The set of all boundary points of S is called the boundary of S, denoted by bd(S).</Paragraph><Paragraph Margin="0,0,0,0" LineHeight="Auto" FontFamily="Trebuchet MS" FontStyle="Normal" FontWeight="Normal" FontSize="24.01" Foreground="#FF000000" xml:space="preserve" /></ListItem><ListItem Margin="38.33,0,0,0" Padding="0,0,0,0" BorderThickness="1,1,1,1"><Paragraph Margin="0,0,0,0" LineHeight="Auto" FontFamily="Trebuchet MS" FontStyle="Normal" FontWeight="Normal" FontSize="24.01" Foreground="#FF000000" xml:space="preserve">A point s in S is called interior point of S if there exists a neighborhood of S completely contained in S. The set of all interior points of S is called the interior, denoted by <LineBreak />int(S).</Paragraph><Paragraph Margin="0,0,0,0" LineHeight="Auto" FontFamily="Trebuchet MS" FontStyle="Normal" FontWeight="Normal" FontSize="24.01" Foreground="#FF000000" xml:space="preserve" /></ListItem><ListItem Margin="38.33,0,0,0" Padding="0,0,0,0" BorderThickness="1,1,1,1"><Paragraph Margin="0,0,0,0" LineHeight="Auto" FontFamily="Trebuchet MS" FontStyle="Normal" FontWeight="Normal" FontSize="24.01" Foreground="#FF000000">A point t in S is called isolated point of S if there exists a neighborhood U of t such that U intersect S = {t}.</Paragraph><Paragraph Margin="0,0,0,0" LineHeight="Auto" FontFamily="Trebuchet MS" FontStyle="Normal" FontWeight="Normal" FontSize="24.01" Foreground="#FF000000" xml:space="preserve" /></ListItem><ListItem Margin="38.33,0,0,0" Padding="0,0,0,0" BorderThickness="1,1,1,1"><Paragraph Margin="0,0,0,0" LineHeight="Auto" FontFamily="Trebuchet MS" FontStyle="Normal" FontWeight="Normal" FontSize="24.01" Foreground="#FF000000">A point r in S is called accumulation point, if every neighborhood of r contains infinitely many distinct points of S.</Paragraph></ListItem></List></FlowDocument>2624719209<FlowDocument FontFamily="Trebuchet MS" FontStyle="Normal" FontWeight="Normal" FontSize="24.01" Foreground="#FF000000" PagePadding="5,0,5,0" AllowDrop="True" NumberSubstitution.CultureSource="User" xmlns="http://schemas.microsoft.com/winfx/2006/xaml/presentation" xmlns:x="http://schemas.microsoft.com/winfx/2006/xaml" xmlns:s="clr-namespace:System;assembly=mscorlib"><Paragraph LineHeight="Auto"><Run FontWeight="Bold">Definition: (Boundary, Accumulation, Interior, and Isolated Points)</Run></Paragraph><Paragraph LineHeight="Auto">Let S be an arbitrary set in the real line R.</Paragraph><Paragraph LineHeight="Auto" /><List MarkerStyle="Disc" MarkerOffset="17" Margin="0,0,0,0" Padding="0,0,0,0" FontFamily="Trebuchet MS" FontStyle="Normal" FontWeight="Normal" FontSize="24.01" Foreground="#FF000000"><List.Tag><s:Int32>0</s:Int32></List.Tag><ListItem Margin="38.33,0,0,0" Padding="0,0,0,0" BorderThickness="1,1,1,1"><Paragraph Margin="0,0,0,0" LineHeight="Auto" FontFamily="Trebuchet MS" FontStyle="Normal" FontWeight="Normal" FontSize="24.01" Foreground="#FF000000">A point b in R is called <Run FontWeight="Bold" xml:space="preserve">boundary point </Run>of S if every non-empty neighborhood of b intersects S and the complement of S. The set of all boundary points of S is called the boundary of S, denoted by bd(S).</Paragraph><Paragraph Margin="0,0,0,0" LineHeight="Auto" FontFamily="Trebuchet MS" FontStyle="Normal" FontWeight="Normal" FontSize="24.01" Foreground="#FF000000" /></ListItem><ListItem Margin="38.33,0,0,0" Padding="0,0,0,0" BorderThickness="1,1,1,1"><Paragraph Margin="0,0,0,0" LineHeight="Auto" FontFamily="Trebuchet MS" FontStyle="Normal" FontWeight="Normal" FontSize="24.01" Foreground="#FF000000" xml:space="preserve">A point s in S is called <Run FontWeight="Bold">interior point </Run>of S if there exists a neighborhood of S completely contained in S. The set of all interior points of S is called the interior, denoted by <LineBreak />int(S).</Paragraph><Paragraph Margin="0,0,0,0" LineHeight="Auto" FontFamily="Trebuchet MS" FontStyle="Normal" FontWeight="Normal" FontSize="24.01" Foreground="#FF000000" /></ListItem><ListItem Margin="38.33,0,0,0" Padding="0,0,0,0" BorderThickness="1,1,1,1"><Paragraph Margin="0,0,0,0" LineHeight="Auto" FontFamily="Trebuchet MS" FontStyle="Normal" FontWeight="Normal" FontSize="24.01" Foreground="#FF000000">A point t in S is called <Run FontWeight="Bold" xml:space="preserve">isolated point </Run>of S if there exists a neighborhood U of t such that U intersect S = {t}.</Paragraph><Paragraph Margin="0,0,0,0" LineHeight="Auto" FontFamily="Trebuchet MS" FontStyle="Normal" FontWeight="Normal" FontSize="24.01" Foreground="#FF000000" /></ListItem><ListItem Margin="38.33,0,0,0" Padding="0,0,0,0" BorderThickness="1,1,1,1"><Paragraph Margin="0,0,0,0" LineHeight="Auto" FontFamily="Trebuchet MS" FontStyle="Normal" FontWeight="Normal" FontSize="24.01" Foreground="#FF000000">A point r in S is called <Run FontWeight="Bold">accumulation point</Run>, if every neighborhood of r contains infinitely many distinct points of S.</Paragraph></ListItem></List></FlowDocument>{\rtf1\ansi\ansicpg1252\uc1\htmautsp\deff2{\fonttbl{\f0\fcharset0 Times New Roman;}{\f2\fcharset0 Trebuchet MS;}}{\colortbl\red0\green0\blue0;\red255\green255\blue255;}\loch\hich\dbch\pard\tx1700\tx3400\tx5100\plain\ltrpar\itap0{\lang1033\fs36\f2\b\cf0 \cf0\ql{\fs40\f2 {\b0\ltrch Example:\tab }\li0\ri0\sa0\sb0\fi0\ql\par} {\fs40\f2 {\b0\ltrch What is the boundary and the interior of (0, 4)?}\li0\ri0\sa0\sb0\fi750\ql\par} {\fs40\f2 {\b0\ltrch Of [-1, 2]?}\li0\ri0\sa0\sb0\fi750\ql\par} {\fs40\f2 {\b0\ltrch Of }{\ltrch R}\li0\ri0\sa0\sb0\fi750\ql\par} {\fs40\f2 {\b0\ltrch And of O ? }\li0\ri0\sa0\sb0\fi750\ql\par} {\fs40\f2 {\b0\ltrch }\li0\ri0\sa0\sb0\fi750\ql\par} {\fs40\f2 {\b0\ltrch Which points are isolated and accumulation points, if any ?}\li0\ri0\sa0\sb0\fi750\ql\par} {\fs40\f2 {\b0\ltrch }\li0\ri0\sa0\sb0\fi0\ql\par} {\fs40\f2 {\b0\ltrch Find the boundary, interior, isolated and accumulation points, if any, for the set \{1, 1/2, 1/3, ... \} union \{0\}}\li0\ri0\sa0\sb0\fi750\ql\par} } }<FlowDocument FontFamily="Trebuchet MS" FontStyle="Normal" FontWeight="Bold" FontSize="24.01" Foreground="#FF000000" PagePadding="5,0,5,0" AllowDrop="True" NumberSubstitution.CultureSource="User" xmlns="http://schemas.microsoft.com/winfx/2006/xaml/presentation"><Paragraph LineHeight="Auto"><Run FontWeight="Normal" FontSize="26.6766666666667" xml:space="preserve">Example: </Run></Paragraph><Paragraph TextIndent="50" LineHeight="Auto"><Run FontWeight="Normal" FontSize="26.6766666666667">What is the boundary and the interior of (0, 4)?</Run></Paragraph><Paragraph TextIndent="50" LineHeight="Auto"><Run FontWeight="Normal" FontSize="26.6766666666667">Of [-1, 2]?</Run></Paragraph><Paragraph TextIndent="50" LineHeight="Auto"><Run FontWeight="Normal" FontSize="26.6766666666667" xml:space="preserve">Of </Run><Run FontSize="26.6766666666667">R</Run></Paragraph><Paragraph TextIndent="50" LineHeight="Auto"><Run FontWeight="Normal" FontSize="26.6766666666667" xml:space="preserve">And of O ? </Run></Paragraph><Paragraph TextIndent="50" LineHeight="Auto"><Run FontWeight="Normal" FontSize="26.6766666666667" xml:space="preserve" /></Paragraph><Paragraph TextIndent="50" LineHeight="Auto"><Run FontWeight="Normal" FontSize="26.6766666666667">Which points are isolated and accumulation points, if any ?</Run></Paragraph><Paragraph LineHeight="Auto"><Run FontWeight="Normal" FontSize="26.6766666666667" xml:space="preserve"> </Run></Paragraph><Paragraph TextIndent="50" LineHeight="Auto"><Run FontWeight="Normal" FontSize="26.6766666666667">Find the boundary, interior, isolated and accumulation points, if any, for the set {1, 1/2, 1/3, ... } union {0}</Run></Paragraph></FlowDocument>2624719209<FlowDocument FontFamily="Trebuchet MS" FontStyle="Normal" FontWeight="Bold" FontSize="24.01" Foreground="#FF000000" PagePadding="5,0,5,0" AllowDrop="True" NumberSubstitution.CultureSource="User" xmlns="http://schemas.microsoft.com/winfx/2006/xaml/presentation"><Paragraph LineHeight="Auto"><Run FontSize="26.6766666666667">Example</Run><Run FontWeight="Normal" FontSize="26.6766666666667">:</Run></Paragraph><Paragraph LineHeight="Auto"><Run FontWeight="Normal" FontSize="26.6766666666667" xml:space="preserve"> </Run></Paragraph><Paragraph TextIndent="50" LineHeight="Auto"><Run FontWeight="Normal" FontSize="26.6766666666667">What is the boundary and the interior of (0, 4)?</Run></Paragraph><Paragraph TextIndent="50" LineHeight="Auto"><Run FontWeight="Normal" FontSize="26.6766666666667" xml:space="preserve" /></Paragraph><Paragraph TextIndent="50" LineHeight="Auto"><Run FontWeight="Normal" FontSize="26.6766666666667">Of [-1, 2]?</Run></Paragraph><Paragraph TextIndent="50" LineHeight="Auto"><Run FontWeight="Normal" FontSize="26.6766666666667" xml:space="preserve" /></Paragraph><Paragraph TextIndent="50" LineHeight="Auto"><Run FontWeight="Normal" FontSize="26.6766666666667" xml:space="preserve">Of </Run><Run FontSize="26.6766666666667">R</Run></Paragraph><Paragraph TextIndent="50" LineHeight="Auto"><Run FontWeight="Normal" FontSize="26.6766666666667" xml:space="preserve" /></Paragraph><Paragraph TextIndent="50" LineHeight="Auto"><Run FontWeight="Normal" FontSize="26.6766666666667" xml:space="preserve">Of O ? </Run></Paragraph><Paragraph TextIndent="50" LineHeight="Auto"><Run FontWeight="Normal" FontSize="26.6766666666667" xml:space="preserve" /></Paragraph><Paragraph TextIndent="50" LineHeight="Auto"><Run FontWeight="Normal" FontSize="26.6766666666667">Which points are isolated and accumulation points, if any ?</Run></Paragraph><Paragraph LineHeight="Auto"><Run FontWeight="Normal" FontSize="26.6766666666667" xml:space="preserve"> </Run></Paragraph><Paragraph TextIndent="50" LineHeight="Auto"><Run FontWeight="Normal" FontSize="26.6766666666667">Find the boundary, interior, isolated and accumulation points, if any, for the set {1, 1/2, 1/3, ... } union {0}</Run></Paragraph></FlowDocument><FlowDocument FontFamily="Trebuchet MS" FontStyle="Normal" FontWeight="Bold" FontSize="24.01" Foreground="#FF000000" PagePadding="5,0,5,0" AllowDrop="True" NumberSubstitution.CultureSource="User" xmlns="http://schemas.microsoft.com/winfx/2006/xaml/presentation" xmlns:x="http://schemas.microsoft.com/winfx/2006/xaml"><Paragraph LineHeight="Auto"><Run FontSize="26.6766666666667">Example</Run><Run FontWeight="Normal" FontSize="26.6766666666667">:</Run></Paragraph><Paragraph LineHeight="Auto"><Run FontWeight="Normal" FontSize="26.6766666666667" xml:space="preserve"> </Run></Paragraph><Paragraph TextIndent="50" LineHeight="Auto"><Run FontWeight="Normal" FontSize="26.6766666666667">What is the boundary and the interior of (0, 4)?</Run></Paragraph><Paragraph TextIndent="50" LineHeight="Auto"><Run FontWeight="Normal" FontSize="26.6766666666667" xml:space="preserve" /></Paragraph><Paragraph TextIndent="50" LineHeight="Auto"><Run FontWeight="Normal" FontSize="26.6766666666667">Of [-1, 2]?</Run></Paragraph><Paragraph TextIndent="50" LineHeight="Auto"><Run FontWeight="Normal" FontSize="26.6766666666667" xml:space="preserve" /></Paragraph><Paragraph TextIndent="50" LineHeight="Auto"><Run FontWeight="Normal" FontSize="26.6766666666667" xml:space="preserve">Of </Run><Run FontSize="26.6766666666667">R</Run></Paragraph><Paragraph TextIndent="50" LineHeight="Auto"><Run FontWeight="Normal" FontSize="26.6766666666667" xml:space="preserve" /></Paragraph><Paragraph TextIndent="50" LineHeight="Auto"><Run FontWeight="Normal" FontSize="26.6766666666667" xml:space="preserve">Of O ? </Run></Paragraph><Paragraph TextIndent="50" LineHeight="Auto"><Run FontWeight="Normal" FontSize="26.6766666666667" xml:space="preserve" /></Paragraph><Paragraph TextIndent="50" LineHeight="Auto"><Run FontWeight="Normal" FontSize="26.6766666666667">Which points are isolated and accumulation points, if any ?</Run></Paragraph><Paragraph LineHeight="Auto"><Run FontWeight="Normal" FontSize="26.6766666666667" xml:space="preserve"> </Run></Paragraph><Paragraph LineHeight="Auto"><Run FontWeight="Normal" FontSize="26.6766666666667" xml:space="preserve" /></Paragraph><Paragraph LineHeight="Auto"><Run FontWeight="Normal" FontSize="26.6766666666667" xml:space="preserve" /></Paragraph><Paragraph TextIndent="50" LineHeight="Auto"><Run FontWeight="Normal" FontSize="26.6766666666667">Find the boundary, interior, isolated and accumulation points, if any, for the set {1, 1/2, 1/3, ... } union {0}</Run></Paragraph></FlowDocument>{\rtf1\ansi\ansicpg1252\uc1\htmautsp\deff2{\fonttbl{\f0\fcharset0 Times New Roman;}{\f2\fcharset0 Trebuchet MS;}}{\colortbl\red0\green0\blue0;\red255\green255\blue255;}\loch\hich\dbch\pard\tx1700\tx3400\tx5100\plain\ltrpar\itap0{\lang1033\fs36\f2\b\cf0 \cf0\ql{\fs32\f2 {\ltrch Proposition: (Boundary, Accumulation, Interior, and Isolated Points)}\li0\ri0\sa0\sb0\fi0\ql\par} {\fs32\f2 {\b0\ltrch \tab }\li0\ri0\sa0\sb0\fi0\ql\par} {\fs32\f2 {\b0\ltrch Let S R. Then each point of S is either an interior point or a boundary point.}\li0\ri0\sa0\sb0\fi0\ql\par} {\fs32\f2 {\b0\ltrch Let S R. Then bd(S) = bd(R \\ S).}\li0\ri0\sa0\sb0\fi0\ql\par} {\fs32\f2 {\b0\ltrch A closed set contains all of its boundary points. An open set contains none of its boundary points.}\li0\ri0\sa0\sb0\fi0\ql\par} {\fs32\f2 {\b0\ltrch Every non-isolated boundary point of a set S R is an accumulation point of S.}\li0\ri0\sa0\sb0\fi0\ql\par} {\fs32\f2 {\b0\ltrch An accumulation point is never an isolated point.}\li0\ri0\sa0\sb0\fi0\ql\par} } }<FlowDocument FontFamily="Trebuchet MS" FontStyle="Normal" FontWeight="Bold" FontSize="24.01" Foreground="#FF000000" PagePadding="5,0,5,0" AllowDrop="True" NumberSubstitution.CultureSource="User" xmlns="http://schemas.microsoft.com/winfx/2006/xaml/presentation"><Paragraph LineHeight="Auto"><Run FontSize="21.3433333333333">Proposition: (Boundary, Accumulation, Interior, and Isolated Points)</Run></Paragraph><Paragraph LineHeight="Auto"><Run FontWeight="Normal" FontSize="21.3433333333333" xml:space="preserve"> </Run></Paragraph><Paragraph LineHeight="Auto"><Run FontWeight="Normal" FontSize="21.3433333333333" xml:space="preserve">Let S R. Then each point of S is either an interior point or a boundary point.</Run></Paragraph><Paragraph LineHeight="Auto"><Run FontWeight="Normal" FontSize="21.3433333333333" xml:space="preserve">Let S R. Then bd(S) = bd(R \ S).</Run></Paragraph><Paragraph LineHeight="Auto"><Run FontWeight="Normal" FontSize="21.3433333333333">A closed set contains all of its boundary points. An open set contains none of its boundary points.</Run></Paragraph><Paragraph LineHeight="Auto"><Run FontWeight="Normal" FontSize="21.3433333333333" xml:space="preserve">Every non-isolated boundary point of a set S R is an accumulation point of S.</Run></Paragraph><Paragraph LineHeight="Auto"><Run FontWeight="Normal" FontSize="21.3433333333333">An accumulation point is never an isolated point.</Run></Paragraph></FlowDocument>2624719209<FlowDocument FontFamily="Trebuchet MS" FontStyle="Normal" FontWeight="Bold" FontSize="24.01" Foreground="#FF000000" PagePadding="5,0,5,0" AllowDrop="True" NumberSubstitution.CultureSource="User" xmlns="http://schemas.microsoft.com/winfx/2006/xaml/presentation" xmlns:s="clr-namespace:System;assembly=mscorlib"><Paragraph LineHeight="Auto">Proposition: (Boundary, Accumulation, Interior, and Isolated Points)</Paragraph><Paragraph LineHeight="Auto"><Run FontWeight="Normal" xml:space="preserve"> </Run></Paragraph><List MarkerStyle="Disc" MarkerOffset="17" Margin="0,0,0,0" Padding="0,0,0,0" FontFamily="Trebuchet MS" FontStyle="Normal" FontWeight="Normal" FontSize="24.01" Foreground="#FF000000"><List.Tag><s:Int32>0</s:Int32></List.Tag><ListItem Margin="38.33,0,0,0" Padding="0,0,0,0" BorderThickness="1,1,1,1"><Paragraph Margin="0,0,0,0" LineHeight="Auto" FontFamily="Trebuchet MS" FontStyle="Normal" FontWeight="Bold" FontSize="24.01" Foreground="#FF000000"><Run FontWeight="Normal">Let S be subset of R. Then each point of S is either an interior point or a boundary point.</Run></Paragraph><Paragraph Margin="0,0,0,0" LineHeight="Auto" FontFamily="Trebuchet MS" FontStyle="Normal" FontWeight="Bold" FontSize="24.01" Foreground="#FF000000"><Run FontWeight="Normal" xml:space="preserve" /></Paragraph></ListItem><ListItem Margin="38.33,0,0,0" Padding="0,0,0,0" BorderThickness="1,1,1,1"><Paragraph Margin="0,0,0,0" LineHeight="Auto" FontFamily="Trebuchet MS" FontStyle="Normal" FontWeight="Bold" FontSize="24.01" Foreground="#FF000000"><Run FontWeight="Normal">Let S be subset of R. Then bd(S) = bd(R \ S).</Run></Paragraph><Paragraph Margin="0,0,0,0" LineHeight="Auto" FontFamily="Trebuchet MS" FontStyle="Normal" FontWeight="Bold" FontSize="24.01" Foreground="#FF000000"><Run FontWeight="Normal" xml:space="preserve" /></Paragraph></ListItem><ListItem Margin="38.33,0,0,0" Padding="0,0,0,0" BorderThickness="1,1,1,1"><Paragraph Margin="0,0,0,0" LineHeight="Auto" FontFamily="Trebuchet MS" FontStyle="Normal" FontWeight="Bold" FontSize="24.01" Foreground="#FF000000"><Run FontWeight="Normal">A closed set contains all of its boundary points. An open set contains none of its boundary points.</Run></Paragraph><Paragraph Margin="0,0,0,0" LineHeight="Auto" FontFamily="Trebuchet MS" FontStyle="Normal" FontWeight="Bold" FontSize="24.01" Foreground="#FF000000"><Run FontWeight="Normal" xml:space="preserve" /></Paragraph></ListItem><ListItem Margin="38.33,0,0,0" Padding="0,0,0,0" BorderThickness="1,1,1,1"><Paragraph Margin="0,0,0,0" LineHeight="Auto" FontFamily="Trebuchet MS" FontStyle="Normal" FontWeight="Bold" FontSize="24.01" Foreground="#FF000000"><Run FontWeight="Normal" xml:space="preserve">Every non-isolated boundary point of a set S R is an accumulation point of S.</Run></Paragraph><Paragraph Margin="0,0,0,0" LineHeight="Auto" FontFamily="Trebuchet MS" FontStyle="Normal" FontWeight="Bold" FontSize="24.01" Foreground="#FF000000"><Run FontWeight="Normal" xml:space="preserve" /></Paragraph></ListItem><ListItem Margin="38.33,0,0,0" Padding="0,0,0,0" BorderThickness="1,1,1,1"><Paragraph Margin="0,0,0,0" LineHeight="Auto" FontFamily="Trebuchet MS" FontStyle="Normal" FontWeight="Bold" FontSize="24.01" Foreground="#FF000000"><Run FontWeight="Normal">An accumulation point is never an isolated point.</Run></Paragraph></ListItem></List></FlowDocument>{\rtf1\ansi\ansicpg1252\uc1\htmautsp\deff2{\fonttbl{\f0\fcharset0 Times New Roman;}{\f2\fcharset0 Trebuchet MS;}}{\colortbl\red0\green0\blue0;\red255\green255\blue255;}\loch\hich\dbch\pard\tx1700\tx3400\tx5100\plain\ltrpar\itap0{\lang1033\fs36\f2\cf0 \cf0\ql{\f2 {\ltrch Theorem: If }\li0\ri0\sa0\sb0\fi0\ql\par} } }<FlowDocument FontFamily="Trebuchet MS" FontStyle="Normal" FontWeight="Normal" FontSize="24.01" Foreground="#FF000000" PagePadding="5,0,5,0" AllowDrop="True" NumberSubstitution.CultureSource="User" xmlns="http://schemas.microsoft.com/winfx/2006/xaml/presentation"><Paragraph xml:space="preserve">Theorem: If </Paragraph></FlowDocument>66692004<FlowDocument FontFamily="Trebuchet MS" FontStyle="Normal" FontWeight="Normal" FontSize="24.01" Foreground="#FF000000" PagePadding="5,0,5,0" AllowDrop="True" NumberSubstitution.CultureSource="User" xmlns="http://schemas.microsoft.com/winfx/2006/xaml/presentation"><Paragraph><Run FontWeight="Bold">Theorem</Run>: A point c is an accumulation point of a set S if and only if there exists a sequence a_n that converges to c.</Paragraph></FlowDocument><FlowDocument FontFamily="Trebuchet MS" FontStyle="Normal" FontWeight="Normal" FontSize="24.01" Foreground="#FF000000" PagePadding="5,0,5,0" AllowDrop="True" NumberSubstitution.CultureSource="User" xmlns="http://schemas.microsoft.com/winfx/2006/xaml/presentation"><Paragraph><Run FontWeight="Bold">Theorem</Run>: A point c is an accumulation point of a set S if and only if there exists a sequence a_n of points in S that converges to c.</Paragraph></FlowDocument><FlowDocument FontFamily="Trebuchet MS" FontStyle="Normal" FontWeight="Normal" FontSize="24.01" Foreground="#FF000000" PagePadding="5,0,5,0" AllowDrop="True" NumberSubstitution.CultureSource="User" xmlns="http://schemas.microsoft.com/winfx/2006/xaml/presentation"><Paragraph><Run FontWeight="Bold">Theorem</Run>: A point c is an accumulation point of a set S if and only if there exists a sequence a_n of points in S that converges to c.</Paragraph></FlowDocument><FlowDocument FontFamily="Trebuchet MS" FontStyle="Normal" FontWeight="Normal" FontSize="24.01" Foreground="#FF000000" PagePadding="5,0,5,0" AllowDrop="True" NumberSubstitution.CultureSource="User" xmlns="http://schemas.microsoft.com/winfx/2006/xaml/presentation"><Paragraph><Run FontWeight="Bold">Theorem</Run>: A point c is an accumulation point of a set S if and only if there exists a sequence a_n of points in S that converges to 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eac83b4f-4eae-4100-9c4c-7c2fa98ccbd6d42fdf07-15ea-452c-ba06-0d503deeec9084f739cb-268f-4bb2-9582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