Definition:Compact Space

From ProofWiki
Jump to navigation Jump to search

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

Definition:Compact Space/Complex Analysis


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:

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


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.