# properties of the closure operator

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

1. 1.

$\overline{A}=A\cup A^{\prime}$ where $A^{\prime}$ is the derived set of $A$.

2. 2.

$A\subseteq\overline{A}$, and $A=\overline{A}$ if and only if $A$ is closed

3. 3.

$\overline{A}=\emptyset$ if and only if $A=\emptyset$.

4. 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 $f(\overline{A})=\overline{f(A)}$.

Title properties of the closure operator PropertiesOfTheClosureOperator 2013-03-22 15:17:05 2013-03-22 15:17:05 matte (1858) matte (1858) 11 matte (1858) Theorem msc 54A99