Letbe the product space of the countable family of non - empty spaces
Thenis compact if and only only each component space is compact.
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.