Topological closure

Let $(X, \tau)$ - a topological-space.

Topological closure of $S \subset X$ is defined as $$ [S]{\tau} = \bigcap{X\setminus F \in \tau, , S \subset F} F. $$

(i.e. a smallest closed-set in terms of inclusion containing $S$)

Equivalent definition

• $[S]_{\tau}$ = {$x \in X$ | $x$ - point-of-closure}

  Prove by inclusion in both directions.