# Definition:Compact Space

## 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 line considered as a topological space under the Euclidean topology.

Let $H \subseteq \R$.

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

### Complex Analysis

Let $D$ be a subset of the complex plane $\C$.

Then $D$ is **compact (in $\C$)** if and only if:

- $D$ is closed in $\C$

and

- $D$ is bounded in $\C$.

### Topology

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

### 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 $K \subseteq X$.

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

That is, if:

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

## 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**.

## Motivation

The question is asked:

*What is compactness useful for?*

to which the following answer can be presented:

- $(1): \quad$ Local properties can be extended to being global properties.

- $(2): \quad$ Compactness allows us to establish properties about a mapping, in particular continuity, in a context of finiteness.

In particular it can be noted that many statements about a mapping $f : A \to B$ are:

- $\text {(a)}: \quad$ Trivially true when $f$ is a finite set

- $\text {(b)}: \quad$ true when $f$ is a continuous mapping when $A$ is a compact space

- $\text {(c)}: \quad$ false, or very difficult to prove when $f$ is a continuous mapping but when $A$ is not compact.

## Also see

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

## Historical Note

Hermann Weyl is reported as having made the observation:

*If a city is compact, it can be guarded by a finite number of arbitrarily near-sighted policemen.*

The specific source of this quote is the subject of research.