Compact

is compact if every open cover of has a finite subcover.

Compact-closed

Theorem

If is compact and is Continuous then is Bounded and attains its bound.

Proof sketch

Consider , a cover of . It has a finite subcover, and is bounded in each set of the subcover, so is bounded on the union of those finitely many sets. Now let and suppose for every we have . Hence find s.t. . Now , so these sets build a cover of . Hence find a finite subcover and smallest from that subcover. But then so:

Theorems

Closed interval is compact Topological Inverse Function Theorem Tychonorff’s Theorem on Finite Products Heine-Borel Theorem