## 5.2. Compact and Perfect Sets

### Examples 5.2.7(a):

Consider the collection of sets

The intersection of all intervals *(0, 1/j)*for all*j > 0*. What is the intersection of all of these sets ?*(0, 1/j)*is empty. To see this, take any real number

*x*. If

*x 0*it is not in any of the intervals

*(0, 1/j)*, and hence not in their intersection. If

*x > 0*, then there exists an integer

*N*such that

*0 < 1 / N < x*. But then

*x*is not in the set

*(0, 1 / N)*and therefore

*x*is not in the intersection. Therefore, the intersection is empty.

Note that this is an intersection of 'nested' sets, that is sets that are decreasing: every 'next' set is a subset of its predecessor.