Definition:Closed Set
This page is about closed sets in the context of Topology (including its application to Metric Spaces, Complex Analysis and Real Analysis). For other uses, see Definition:Closed.
Contents
Definition
Topology
Let $T = \left({S, \tau}\right)$ 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 $\left({S \setminus H}\right) \in \tau$.
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 $F \subset X$.
$F$ is closed in $V$ if and only if its complement $X \setminus F$ is open in $V$.
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) |