# Definition:Closed Set

## 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 $\cl: \powerset S \to \powerset S$ be a closure operator.

Let $T \subseteq S$ be a subset.

### Definition 1

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

$\map \cl T = T$

### Definition 2

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

$T \in \Img \cl$

## Also see

• Results about Closed Sets can be found here.

## Internationalization

Closed (in this context) is translated:

 In French: fermé (literally: closed)