# 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 $T = \left({S, \tau}\right)$ is compact if and only if every filter on $S$ has a limit point in $S$.

### Definition 5

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

## Also defined as

A compact topological space is often additionally required to be Hausdorff. What is called a compact space here is then called a quasicompact space.

## Also see

• Results about compact spaces can be found here.