Definition:Closure (Topology)
This page is about closure in the context of Topology. For other uses, see closure.
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.