Compactness and Product Topologies

So, I have a problem that I'm working on and I can't seem to figure it out. We just started product topologies, so many properties are still new or unknown to me. Here's the problem:

Let and be topological spaces. If is compact relative to and , show that is compact relative to the product topology on .

I'm not seeing why being compact is sufficient for the whole product to be compact... Any help would be appreciated.

Re: Compactness and Product Topologies

Quote:

Originally Posted by

**Aryth** Let

and

be topological spaces. If

is compact relative to

and

, show that

is compact relative to the product topology on

.

I'm not seeing why

being compact is sufficient for the whole product to be compact... Any help would be appreciated.

Are you sure that you have copied the question correctly?

Could it be show that is compact relative to the product topology on .

Re: Compactness and Product Topologies

Quote:

Originally Posted by

**Plato** Are you sure that you have copied the question correctly?

Could it be show that

is compact relative to the product topology on

.

If it is then I certainly understand it, but the question definitely says . I'll ask my professor tomorrow if the problem as stated has a typo and prove the revision instead. Thanks for the help.