Compact-closed

Let . Then:

1. If is compact and is closed in , then is compact
2. If is Hausdorff and is compact, then is closed

Proof sketch

1. Let be an open cover of . For each , find open in s.t. , and name this collection . Then is an open cover of . Find a finite subcover of . Intersect with to get back a finite subcover of .
2. We show is open. Let . For any find open and disjoint s.t. and (because is Hausdorff). Now, certainly makes an open cover of , so find a finite subcover and let be the finite set of s s.t. makes this finite subcover. Then take . This is an open nbd of disjoint from .