1.3. Basic Topology

Theorem 1.3.X: Triangle Inequality

For all complex numbers z and w we have:
|z + w| |z| + |w|

blah blah

Definition 1.3.x: Disks, Open, and Closed Sets

The set D(z₀, r) is the open disk centered at z₀ with radius r. In other words:
D(z₀, r) = {z C : |z - z₀| < r }
A set U is called open if for every z U there exists an r > 0 such that D(z, r) U. A set C is called closed if its complement comp(C) is open.

blah blah

Example 1.3.X: Simple sets inC

If c is a (complex) constant, then clearly the set {z = c} is a single point in the plane. Describe the sets

{ Re(z) = c}

{ Im(z) = c}

{ Arg(z) = c}

{ |z| = c}

{ z = }
{ r cis(t) = c}

{ z = r cis(t): r = c}

{ z = r cis(t): t = c}

blah blah

Example 1.3.X: Disks and friends

The set { |z - c| < r } is an open disk centered at c with radius r (explain). What are:

{ |z - c| > r }

{ |z - c| r }

{ r₁ < |z - c| < r₂ }

{ c₁ < Arg(z) < c₂ }

{ |z - 2z| < r}

circles, disks (inside/outside), annulus, lines, half-planes

Interactive Complex Analysis, ver. 1.0.0
(c) 2006-2007, Bert G. Wachsmuth
Page last modified: May 29, 2007

*Theorem 1.3.X: Triangle Inequality*
	For all complex numbers z and w we have: \|z + w\| \|z\| + \|w\|

*Definition 1.3.x: Disks, Open, and Closed Sets*
	The set D(z₀, r) is the open disk centered at z₀ with radius r. In other words: D(z₀, r) = {z C : \|z - z₀\| < r } A set U is called open if for every z U there exists an r > 0 such that D(z, r) U. A set C is called closed if its complement comp(C) is open.

*Example 1.3.X: Simple sets inC*
	If c is a (complex) constant, then clearly the set {z = c} is a single point in the plane. Describe the sets { Re(z) = c} { Im(z) = c} { Arg(z) = c} { \|z\| = c} { z = } { r cis(t) = c} { z = r cis(t): r = c} { z = r cis(t): t = c}

*Example 1.3.X: Disks and friends*
	The set { \|z - c\| < r } is an open disk centered at c with radius r (explain). What are: { \|z - c\| > r } { \|z - c\| r } { r₁ < \|z - c\| < r₂ } { c₁ < Arg(z) < c₂ } { \|z - 2z\| < r}