Definition:Closure (Topology)

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

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

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 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.