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 $M = \struct {A, d}$ be a metric space.
Let $\tau$ denote the topology on $A$ induced by $d$.
Then $M$ is compact if and only if $\struct {A, \tau}$ 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.