Definition:Closure (Topology)

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

Then the closure of $$A$$, denoted by $$\overline{A}$$ or $$\operatorname{cl}(A)$$, is defined as
 * $$\overline{A} := \bigcap_{A \subseteq B \subseteq X, B \text{ closed}} B$$.

Equivalently,
 * $$\overline{A}$$ is the smallest closed set that contains $$A$$;
 * $$\overline{A}$$ is the union of $$A$$ and its boundary;
 * $$\overline{A}$$ is the union of $$A$$ and its limit points;
 * $$\overline{A}$$ is the union of all isolated points of $$A$$ and all limit points of $$A$$.

This is demonstrated in Equivalent Definitions for Topological Closure.