Compact space

Definition
A topological space is termed a compact space if it satisfies the following equivalent conditions:


 * Every open cover of the topological space has a finite subcover.
 * If a collection of closed subsets of the space has the property that every finite subcollection has a nonempty intersection, then the whole collection has a nonempty intersection.
 * The only maximal ideals in the ring of continuous real-valued functions on the topological space are the ideals of functions vanishing at a particular point.

Equivalence of definitions
The third definition is equivalent to the first two because: Natural map from topological space to max-spectrum of ring of continuous real-valued functions is a surjection iff the space is compact.

Primary subject wiki entry
Topospaces:Compact space