Definition:Closure (Topology)

Definition
Let $T$ be a topological space, and let $H \subseteq T$.

Then the closure of $H$ is defined as:
 * $\operatorname{cl} \left({H}\right)$ is the union of $H$ and its limit points.

From the definition of derived set, this is equivalent to:
 * $\operatorname{cl} \left({H}\right) = H \cup H'$

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

Notation
The 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 ambiguity problems 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 regarded by some as cumbersome, but they have the advantage of being clear.

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

On this website, $\operatorname{cl} \left({H}\right)$ and $H^-$ are the notations of choice.

Equivalent Definitions
The following definitions for closure are equivalent to the above:


 * $\displaystyle H^- := \bigcap_{H \subseteq K \subseteq T, K \text{ closed}} K$
 * $H^-$ is the smallest closed set that contains $H$
 * $H^-$ is the union of $H$ and its boundary
 * $H^-$ is the union of all isolated points of $H$ and all limit points of $H$.

This fact is demonstrated in Equivalent Definitions for Topological Closure.

Also see

 * Interior