Examples 5.2.5(b):
Let S = [0, 1]. Define
=
{ t R :
| t - | <
and
S}
for a fixed > 0.
Is the collection of all
{ },
S,
an open cover for S ? How many sets of type
are actually needed to cover S ?
First, each set
is an open set, because it is the same as an interval around
of length
2 .
Second, the union of all sets
equals the open interval
(- ,
1 + ),
so it contains the set S. Therefore, the collection
{ },
S
is an open cover of S.
The collection { }, S consists of uncountable many sets. In order to cover S, however, we need only a finite subcollection for any given . To see this, fix an > 0. Then let N be the smallest integer greater than 1 / , and define
- = k * , k = 0, 1, 2, ... N