Suppose $X$ is a topological space, and let $\overline{A}$ be the closure of $A$ in $X$ . Then the following properties hold:

1. $\overline{A}=A\cup A'$ where $A'$ is the derived set of $A$ .
2. $A\subseteq \overline{A}$ , and $A=\overline{A}$ if and only if $A$ is closed
3. $\overline{A}=\emptyset$ if and only if $A=\emptyset$ .
4. If $Y$ is another topological space, then $f\colon X \to Y$ is a continuous map, if and only if $f(\overline{A}) \subseteq \overline{f(A)}$ for all $A\subseteq X$ . If $f$ is also a homeomorphism, then $f(\overline{A}) = \overline{f(A)}$ .
5. If $E\subseteq X$ is any set, then $$ A\cap \overline{E} \subseteq \overline{A\cap E}. $$