# Definition:Closure (Topology)

## Contents

## Definition

Let $T = \left({S, \tau}\right)$ be a topological space.

Let $H \subseteq S$.

### Definition 1

The **closure of $H$ (in $T$)** is defined as:

- $H^- := H \cup H'$

where $H'$ is the derived set of $H$.

### Definition 2

The **closure of $H$ (in $T$)** is defined as:

- $\displaystyle H^- := \bigcap \left\{{K \supseteq H: K}\right.$ is closed in $\left.{T}\right\}$

### Definition 3

The **closure of $H$ (in $T$)**, denoted $H^-$, is defined as the smallest closed set of $T$ that contains $H$.

### Definition 4

The **closure of $H$ (in $T$)** is defined as the union of $H$ and its boundary in $T$:

- $H^- := H \cup \partial H$

### Definition 5

The **closure of $H$ (in $T$)** is the union of the set of all isolated points of $H$ and the set of all limit points of $H$:

- $H^- := H^i \cup H'$

## Notation

The topological closure of $H$ is variously denoted:

- $\operatorname{cl} \left({H}\right)$
- $\operatorname{Cl} \left({H}\right)$
- $\overline H$
- $H^-$

Of these, it can be argued that $\overline H$ has more problems with ambiguity than the others, as it is also frequently used for the set complement.

$\operatorname{cl} \left({H}\right)$ and $\operatorname{Cl} \left({H}\right)$ are somewhat cumbersome, but they have the advantage of being clear.

$H^-$ is neat and compact, but has the disadvantage of being relatively obscure.

On $\mathsf{Pr} \infty \mathsf{fWiki}$, $H^-$ is notation of choice, although $\operatorname{cl} \left({H}\right)$ can also be found in places.

## Also see

- Results about
**set closures**can be found here.