## 5.2. Compact and Perfect Sets

### Examples 5.2.5(c):

*C = { (1/j, 1)*, for all

*j > 0*}. Is C an open cover for S ? How many sets from the collection C are actually needed to cover S ?

**C**is open. Also

*(1 / j, 1) = (0, 1)*

**C**is indeed an open cover of

**S**.

Not all sets from the original collection **C** are needed to
cover **S**. For example, the subcollection of intervals
*(1 / (2 j) , 1)* is also an open covering. However, we
can not reduce this cover to a finite subcovering. To see this,
extract finitely many sets of the form *(1 / j , 1)* from the
collection **C**. Let *N* be the largest integer *j*
that occurs in this subcollection. Then the point
*1 / (N + 1)* is in **S**, but it is not in any of the
intervals of the finite subcollection. Hence, no finite
subcollection from **C** can cover **S**.

On the other hand, **S** does have some other finite open
coverings. For example, the collection
*{ (-1, 1/2), (0, 2) }* is such a finite open cover. However,
the giving open covering **C** from above can not be reduced to a
finite subcover.