# Definition:Compact Space/Topology

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## Contents

## 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.

## Sources

- 1975: W.A. Sutherland:
*Introduction to Metric and Topological Spaces*... (previous) ... (next): $5.2$: Definition of compactness: Definition $5.2.2$