PropertiesOfTheClosureOperator 2005-05-18 14:23:08 2006-10-16 10:08:04 Theorem % this is the default PlanetMath preamble. as your knowledge % of TeX increases, you will probably want to edit this, but % it should be fine as is for beginners. % almost certainly you want these \usepackage{amssymb} \usepackage{amsmath} \usepackage{amsfonts} \usepackage{amsthm} \usepackage{mathrsfs} % used for TeXing text within eps files %\usepackage{psfrag} % need this for including graphics (\includegraphics) %\usepackage{graphicx} % for neatly defining theorems and propositions % % making logically defined graphics %\usepackage{xypic} % there are many more packages, add them here as you need them % define commands here \newcommand{\sR}[0]{\mathbb{R}} \newcommand{\sC}[0]{\mathbb{C}} \newcommand{\sN}[0]{\mathbb{N}} \newcommand{\sZ}[0]{\mathbb{Z}} \usepackage{bbm} \newcommand{\Z}{\mathbbmss{Z}} \newcommand{\C}{\mathbbmss{C}} \newcommand{\F}{\mathbbmss{F}} \newcommand{\R}{\mathbbmss{R}} \newcommand{\Q}{\mathbbmss{Q}} \newcommand*{\norm}[1]{\lVert #1 \rVert} \newcommand*{\abs}[1]{| #1 |} \newtheorem{thm}{Theorem} \newtheorem{defn}{Definition} \newtheorem{prop}{Proposition} \newtheorem{lemma}{Lemma} \newtheorem{cor}{Corollary} Suppose $X$ is a topological space, and let $\overline{A}$ be the closure of $A$ in $X$. Then the following properties hold: \begin{enumerate} \item $\overline{A}=A\cup A'$ where $A'$ is the derived set of $A$. \item $A\subseteq \overline{A}$, and $A=\overline{A}$ if and only if $A$ is closed \item $\overline{A}=\emptyset$ if and only if $A=\emptyset$. \item 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 $f(\overline{A}) = \overline{f(A)}$. \item If $E\subseteq X$ is any set, then $$ A\cap \overline{E} \subseteq \overline{A\cap E}. $$ \end{enumerate}