The Closure is the Smallest Closed Superset

Let $(X,\mathcal{T})$ be a topological space, and let $A\subseteq X$. Let $\operatorname{cl}_X(A)$ denote the closure of $A$ in $X$, and call a subset of $X$ closed when it is closed in the topological space $(X,\mathcal{T})$.

Then the following hold.

1. $A\subseteq\operatorname{cl}_X(A)$.

2. $\operatorname{cl}_X(A)$ is closed.

3. If $C\subseteq X$ is closed and $A\subseteq C$, then $\operatorname{cl}_X(A)\subseteq C$.

4. $A$ is closed if and only if $A=\operatorname{cl}_X(A)$.

5. If $B\subseteq X$ and $A\subseteq B$, then $\operatorname{cl}_X(A)\subseteq\operatorname{cl}_X(B)$.