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

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

3. $\overline{A}=\mathrm{\varnothing }$ if and only if $A=\mathrm{\varnothing }$.

4. If $Y$ is another topological space, then $f:X\to Y$ 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
Canonical name	PropertiesOfTheClosureOperator
Date of creation	2013-03-22 15:17:05
Last modified on	2013-03-22 15:17:05
Owner	matte (1858)
Last modified by	matte (1858)
Numerical id	11
Author	matte (1858)
Entry type	Theorem
Classification	msc 54A99