Definition:Closure (Topology)

Definition
Let $T$ be a topological space.

Let $H \subseteq T$.

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.

Equivalence of Definitions
The definitions for closure are equivalent.

This is demonstrated in Equivalent Definitions for Topological Closure.

Also see

 * Topological Closure is Closure Operator


 * Interior
 * Boundary