# Definition:Closure (Topology)

## Definition

Let $T = \struct {S, \tau}$ 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:

$\ds H^- := \bigcap \leftset {K \supseteq H: K}$ is closed in $\rightset T$

### 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 $H$:

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

### Definition 6

The closure of $H$ (in $T$), denoted $H^-$, is the set of all adherent points of $H$.

## Metric Space

Let $M = \struct {A, d}$ be a metric space.

Let $H \subseteq A$.

Let $H'$ be the set of limit points of $H$.

Let $H^i$ be the set of isolated points of $H$.

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

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

## Notation

The closure operator of $H$ is variously denoted:

$\map \cl H$
$\map {\mathrm {Cl} } H$
$\overline H$
$H^-$

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.

$\map \cl H$ and $\map {\mathrm {Cl} } H$ are 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 $\map \cl H$ can also be found in places.

## Examples

### Singleton Union with Open Interval

Let $\R$ be the set of real numbers.

Let $H \subseteq \R$ be the subset of $\R$ defined as:

$H = \set 0 \cup \openint 1 2$

Then the closure of $H$ in $\R$ is:

$H^- = \set 0 \cup \closedint 1 2$

### Open Interval in Open Unbounded Interval

Let $S$ be the open real interval:

$S = \openint a \to$

Let $H$ be the open real interval:

$H = \openint a b$

Then the closure of $H$ in $S$ is:

$H^- = \hointl a b$

### Open Interval under Discrete Topology

Let $\T = \struct {\R, \tau_d}$ denote the topological space formed from the set of real numbers $\R$ together with the discrete topology $\tau_d$.

Let $H$ be the open real interval:

$H = \openint a b$

Then the closure of $H$ in $T$ is:

$H^- = \openint a b$

## Also see

• Results about set closures can be found here.