# Interactive Real Analysis

Next | Previous | Glossary | Map

## 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.
Next | Previous | Glossary | Map