54-00-ZeroDimensional.tex

\documentclass[12pt]{article}
\usepackage{pmmeta}
\pmcanonicalname{ZeroDimensional}
\pmcreated{2013-03-22 14:41:05}
\pmmodified{2013-03-22 14:41:05}
\pmowner{matte}{1858}
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\pmtitle{zero dimensional}
\pmrecord{9}{36291}
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\pmtype{Definition}
\pmcomment{trigger rebuild}
\pmclassification{msc}{54-00}
\pmsynonym{zero-dimensional}{ZeroDimensional}
\pmrelated{SeparationAxioms}

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\begin{document}
\begin{defn} \cite{steen, willard}
Suppose $X$ is a topological space. If $X$ has a basis consising of
clopen sets, then $X$ is said to be \PMlinkescapetext{\emph{zero dimensional}}. 
\end{defn}

Examples of zero-dimensional spaces are: the set $\mathbb{Q}$ of rational numbers (with subspace topology induced from the usual metric topology on $\mathbb{R}$, the set of real numbers), the Cantor space, as well as the Sorgenfrey line.

The concepts of zero-dimentionality and total disconnectedness are closely related.  Indeed, every zero-dimentional \PMlinkname{$T_1$ space}{T1Space} is totally disconnected.  Furthermore, if a topological space is locally compact and Hausdorff, then the notions of zero-dimentionality and total disconnectedness are equivalent.


\begin{thebibliography}{9}
\bibitem{steen} L.A. Steen, J.A.Seebach, Jr.,
\emph{Counterexamples in topology},
Holt, Rinehart and Winston, Inc., 1970.
\bibitem{willard} S. Willard, \emph{General Topology},
Addison-Wesley, Publishing Company, 1970.
\end{thebibliography}
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