Most references on topological spaces seem to define a compact space in terms of open coverings of a space having finite subcoverings. Formally, we say that a collection of open sets , indexed by some set , is an open covering of a topological space if . We define to be compact if there exist finitely many open sets in the open covering such that .
There exist a number of equivalent definitions of a compact space, which we briefly mention here. A topological space is compact if and only if has the finite intersection property: if is a collection of a closed sets indexed by a set such that, for any finite subset , , then .
Another equivalent definition is in terms of nets, also known as Moore-Smith sequences. A topological space is compact if and only if every net in has a convergent subnet.
We now prove the following theorem:
A topological space is compact if and only if, for every topological space , the projection map is closed, where is endowed with the product topology.
Here is the proof.