Re: Countably Compact Sets

Quote:

Originally Posted by

**Aryth** Let

be a topological space and let

be countably compact. Prove that every subset of

that is closed relative to

is also countably compact.

As always this depends upon the definitions in use.

If that is closed relative to . Can you show that any countable open cover of has a finite sub cover all relative to

Re: Countably Compact Sets

Ah, so it is similar to the proof with compactness... I'll just change the proof a little, then. If you could verify that it is correct I would grateful.

Suppose that is countably compact and let be closed. We will let be a countable cover of . Since is open, it follows that along with is a countable open cover of . We know that is countably compact, so can be covered by a finite number of sets from , , and possibly . Since , covers . Therefore, is countably compact.

Re: Countably Compact Sets

Quote:

Originally Posted by

**Aryth** Ah, so it is similar to the proof with compactness... I'll just change the proof a little, then. If you could verify that it is correct I would grateful.

Suppose that

is countably compact and let

be closed. We will let

be a countable cover of

. Since

is open, it follows that

along with

is a countable open cover of

. We know that

is countably compact, so

can be covered by a finite number of sets from

,

, and possibly

. Since

,

covers

. Therefore,

is countably compact.

Good. That will work.