Definition A set E is compact if and only if, for every family
Example: Let E=(0,1] and for each positive integer n, let
If we choose a finite set
where
Thus, we have a family of open sets
Heine-Borel Theorom: A set
Examples:
- [2,8] is a compact set.
- The unit disk including the boundary is a compact set.
- (3,5] is not a compact set.
Note that all of these examples are of sets that are uncountably infinite.
Definitions from: Introduction to Analysis 5th edition by Edward D. Gaughan
Theorom: The union of compact sets is compact.