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 .