PseudometricTopology 2004-10-03 06:12:47 2007-06-02 02:49:13 Definition pseudometric space pseudometrizable pseudometric topology pseudo-metric pseudometrizable topological space pseudo-metrizable topological space % 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{\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} Let $(X,d)$ be a pseudometric space. As in a metric space, we define $$ B_\varepsilon(x)=\{ y\in X\mid d(x,y)<\varepsilon \}. $$ for $x\in X$, $\varepsilon>0$. In the below, we show that the collection of sets $$ \mathscr{B}= \{ B_\varepsilon(x)\mid \varepsilon>0, x\in X\} $$ form a base for a topology for $X$. We call this topology the \PMlinkescapetext{\emph{pseudometric topology}} on $X$ induced by $d$. Also, a topological space $X$ is a \emph{pseudometrizable topological space} if there exists a pseudometric $d$ on $X$ whose pseudometric topology coincides with the given topology for $X$ \cite{kelley, willard}. \begin{prop} $\mathscr{B}$ is a base for a topology. \end{prop} \begin{proof} We shall use the \PMlinkid{this result}{5845} to prove that $\mathscr{B}$ is a base. First, as $d(x,x)=0$ for all $x\in X$, it follows that $\mathscr{B}$ is a cover. Second, suppose $B_1,B_2\in \mathscr{B}$ and $z\in B_1\cap B_2$. We claim that there exists a $B_3\in \mathscr{B}$ such that \begin{eqnarray} \label{ii} z&\in& B_3\subseteq B_1\cap B_2. \end{eqnarray} By definition, $B_1 = B_{\varepsilon_1}(x_1)$ and $B_2 = B_{\varepsilon_2}(x_2)$ for some $x_1,x_2\in X$ and $\varepsilon_1,\varepsilon_2>0$. Then $$ d(x_1, z)<\varepsilon_1, \quad d(x_2, z)<\varepsilon_2. $$ Now we can define $\delta = \min\{ \varepsilon_1-d(x_1, z), \varepsilon_2-d(x_2, z)\}>0$, and put $$ B_3 = B_\delta(z). $$ If $y\in B_3$, then for $k=1,2$, we have by the triangle inequality \begin{eqnarray*} d(x_k,y) &\le & d(x_k, z) + d(z,y) \\ &< & d(x_k, z) + \delta \\ &\le & \varepsilon_k, \end{eqnarray*} so $B_3\subseteq B_k$ and condition \ref{ii} holds. \end{proof} \subsubsection*{Remark} In the proof, we have not used the fact that $d$ is symmetric. Therefore, we have, in fact, also shown that any quasimetric induces a topology. \begin{thebibliography}{9} \bibitem{kelley} J.L. Kelley, \emph{General Topology}, D. van Nostrand Company, Inc., 1955. \bibitem{willard} S. Willard, \emph{General Topology}, Addison-Wesley, Publishing Company, 1970. \end{thebibliography}