compact gives:Given topologies and on a set with , prove that if is compact then so is .

Let be any open cover of . Then there exists a finite subcover of , say such that .

Each is open (is the "finite subcover" always open?) so each .

From here I get stuck. If we consider cases, if every then i'm done! However, if then i'm not really sure what to do. I don't even think this is possible since 's are open so should be in anyway.

I'd really appreciate some help!