Closed Set and Closure

Statements

Proposition. (@Bou95, pp. 24) Let $(X,\tau )$ be a topological space, and $S\subseteq X$. Then,

1. $S$ is closed if and only if $\mathrm{cl}(S)=S$. In order words, $S$ is closed if and only if it is equal to is closure.
2. The closure of $S$ is the smallest closed set that contains $S$.

References