## 5.2. Compact and Perfect Sets

### Examples 5.2.5(b):

Let S = [0, 1]. Define

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 *= { 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 ?**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*

*k = 0, 1, 2, ..., N*is a covering of

**S**. That is, this new collection forms a finite subcover of

**S**with respect to the original collection of sets.