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  20130322 15:17:05 
Last modified on  20130322 15:17:05 
Owner  matte (1858) 
Last modified by  matte (1858) 
Numerical id  11 
Author  matte (1858) 
Entry type  Theorem 
Classification  msc 54A99 