Definition:Compact Space

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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.