Definition:Closure (Topology)

Definition
Let $X$ be a topological space, and let $A \subseteq X$.

Then the closure of $A$ is defined as:
 * $\displaystyle \operatorname{cl} \left({A}\right) := \bigcap_{A \subseteq B \subseteq X, B \text{ closed}} B$

That is, it is the intersection of all the closed sets of $X$ which contain $A$.

The closure of $A$ is variously denoted:
 * $\operatorname{cl} \left({A}\right)$
 * $\operatorname{Cl} \left({A}\right)$
 * $\overline A$
 * $A^-$

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

$\operatorname{cl} \left({A}\right)$ and $\operatorname{Cl} \left({A}\right)$ are regarded by some as cumbersome, but they have the advantage of being clear.

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

Equivalent Definitions
The following definitions for closure are equivalent to the above:
 * $A^-$ is the smallest closed set that contains $A$
 * $A^-$ is the union of $A$ and its boundary
 * $A^-$ is the union of $A$ and its limit points
 * $A^-$ is the union of all isolated points of $A$ and all limit points of $A$.

This fact is demonstrated in Equivalent Definitions for Topological Closure.

Also see

 * Interior