Tychonorff's Theorem on Finite Products

Finite products of compact spaces are compact

Proof

Enough to show for product of 2 components.
Let be compact, and let cover .
Fix and for all find s.t. .
Now find open nbds of and of in and respectively
s.t. .
Now cover so find a finite s.t. .
Hence define .
Now will cover so find finite … now cover with these.