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