# Definition:Compact Space

## Contents

## Definition

### Euclidean Space

Let $\R^n$ denote Euclidean $n$-space.

Let $H \subseteq \R^n$.

Then $H$ is **compact in $\R^n$** if and only if $H$ is closed and bounded.

### Real Analysis

The same definition applies when $n = 1$, that is, for the real number line:

Let $\R$ be the real number space considered as a topological space under the Euclidean topology.

Let $H \subseteq \R$.

Then $H$ is **compact in $\R$** if and only if $H$ is closed and bounded.

### Topology

### Definition 1

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

### Definition 2

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

### Definition 3

A topological space $T = \left({S, \tau}\right)$ is **compact** if and only if $\tau$ has a sub-basis $\mathcal B$ such that:

- from every cover of $S$ by elements of $\mathcal B$, 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.

### Metric Space

Let $M = \left({A, d}\right)$ be a metric space.

Let $\tau$ denote the topology on $A$ induced by $d$.

Then $M$ is **compact** if and only if $\left({A, \tau}\right)$ is a compact topological space.

### Normed Vector Space

Let $\struct {X, \norm {\,\cdot\,} }$ be a normed vector space.

Let $K \in X$.

Then $K$ is **compact** if and only if every sequence in $K$ has a convergent subsequence with limit $L \in K$.

That is, if:

- $\sequence {x_n}_{n \mathop \in \N} :\forall n \in \N : x_n \in K \implies \exists \sequence {x_{n_k} }_{k \mathop \in \N} : \exists L \in K: \displaystyle \lim_{k \mathop \to \infty} x_{n_k} = L$

## Also see

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