Definition:Compact Space/Topology

Definition

Definition 1

A topological space $T = \struct {S, \tau}$ is compact if and only if every open cover for $S$ has a finite subcover.

Definition 2

A topological space $T = \struct {S, \tau}$ is compact if and only if it satisfies the Finite Intersection Axiom.

Definition 3

A topological space $T = \struct {S, \tau}$ is compact if and only if $\tau$ has a sub-basis $\BB$ such that:

from every cover of $S$ by elements of $\BB$, a finite subcover of $S$ can be selected.

Definition 4

A topological space $T = \struct {S, \tau}$ is compact if and only if every filter on $S$ has a limit point in $S$.

Definition 5

A topological space $T = \struct {S, \tau}$ is compact if and only if every ultrafilter on $S$ converges.

Also defined as

Some sources, in their definition of a compact space, impose the additional criterion that such a space should also be Hausdorff.

What is called a compact space here is then referred to as a quasicompact (or quasi-compact) space.

Also see

• Results about compact spaces can be found here.