# 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.

### Definition 6

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

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

## 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**.

## Examples

## Examples of Non-Compact Spaces

## Also see

- Results about
**compact spaces**can be found**here**.