Definition:Closed Set

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This page is about Closed Set in the context of topology. For other uses, see Closed.


Definition

Topology

Let $T = \struct {S, \tau}$ be a topological space.

Let $H \subseteq S$.

$H$ is closed (in $T$) if and only if its complement $S \setminus H$ is open in $T$.

That is, $H$ is closed if and only if $\paren {S \setminus H} \in \tau$.

That is, if and only if $S \setminus H$ is an element of the topology of $T$.


Metric Space

In the context of metric spaces, the same definition applies:

$H$ is closed (in $M$) if and only if its complement $A \setminus H$ is open in $M$.


Normed Vector Space

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

Let $F \subset X$.


Definition 1

$F$ is closed in $V$ if and only if its complement $X \setminus F$ is open in $V$.


Definition 2

$F$ is closed (in $V$) if and only if every limit point of $F$ is also a point of $F$.


That is: if and only if $F$ contains all its limit points.


Complex Analysis

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

$S$ is closed (in $\C$) if and only if every limit point of $S$ is also a point of $S$.


That is: if and only if $S$ contains all its limit points.


Real Analysis

Let $S \subseteq \R$ be a subset of the set of real numbers.


Then $S$ is closed (in $\R$) if and only if its complement $\R \setminus S$ is an open set.


Under Closure Operator

The concept of closure can be made more generally than on a topological space:


Let $S$ be a set.

Let $\operatorname{cl}: \mathcal P \left({S}\right) \to \mathcal P \left({S}\right)$ be a closure operator.

Let $T \subseteq S$ be a subset.


Definition 1

The subset $T$ is closed (with respect to $\operatorname{cl}$) if and only if:

$\operatorname{cl} \left({T}\right) = T$


Definition 2

The subset $T$ is closed (with respect to $\operatorname{cl}$) if and only if $T$ is in the image of $\operatorname{cl}$:

$T \in \operatorname{im}(\operatorname{cl})$


Also see

  • Results about Closed Sets can be found here.

Internationalization

Closed (in this context) is translated:

In French: fermé  (literally: closed)