## 5.2. Compact and Perfect Sets

### Example 5.2.10(a):

Find a perfect set. Find a closed set that is not perfect. Find a
compact set that is not perfect. Find an unbounded closed set that
is not perfect. Find a closed set that is neither compact nor
perfect.

- A perfect set needs to be closed, such as the closed interval
*[a, b]*. In fact, every point in that interval*[a, b]*is an accumulation point, so that the set*[a, b]*is a perfect set. - The simplest closed set is a singleton
*{ b }*.The element*b*in then set*{ b }*is not an accumulation point, so the set*{ b }*is closed but not perfect. - The set
*{ b }*from above is also compact, being closed an bounded. Hence, it is compact but not perfect. - The set
*{-1} [0, )*is closed, unbounded, but not perfect, because the element -1 is not an accumulation point of the set. - The set
*{-1} [0, )*from above is closed, not perfect, and also not compact, because it is unbounded.