Tychonoff's Theorem

Theorem

Letbe the product space of the countable family of non - empty spaces

Thenis compact if and only only each component space is compact.

Proof

Supposeis compact. For eachthe projection mapis continuous and onto.

Thereforeis compact for each

Conversely suppose eachis compact. Letbe any sequence inwith the kth component ofwritten asThenis an ultranet inandconverges inThereforeconverges in

A spaceis compact if every sequence inconverges to a point inhenceis compact.