All sets are contained inside their closure

up:: Closure (Topology)

Let $(X,\tau )$ be a Topological Space, and $S\subseteq X$.

Then, by the definition of Closure (Topology), we know that $\overline{S}$ is the intersection of closed sets which enclose $S$.

Since all of them enclose $S$, their intersection will also enclose $S$. The “upper bound” of its closure is $X$ itself, since $X$ is closed; the “lower bound” is $S$ itself, since all closed sets involved enclose $S$; thus, $S\subseteq \overline{S}\subseteq X$.