IndiscreteTopology 2002-06-19 00:41:15 2006-08-22 02:05:45 Definition % 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} % used for TeXing text within eps files %\usepackage{psfrag} % need this for including graphics (\includegraphics) %\usepackage{graphicx} % for neatly defining theorems and propositions %\usepackage{amsthm} % making logically defined graphics %\usepackage{xypic} % there are many more packages, add them here as you need them % define commands here If $X$ is a set and it is endowed with a topology defined by $$\tau=\{X,\emptyset\} \label{eq12}$$ then $X$ is said to have the \emph{indiscrete topology}. Furthermore $\tau$ is the coarsest topology a set can possess, since $\tau$ would be a subset of any other possible topology. This topology gives $X$ many properties: \begin{itemize} \item Every subset of $X$ is sequentially compact. \item Every function to a space with the indiscrete topology is continuous. \item $X$ is path connected and hence connected but is arc connected only if $X$ is uncountable or if $X$ has at most a single point. However, $X$ is both hyperconnected and ultraconnected. \item If $X$ has more than one point, it is not metrizable because it is not Hausdorff. However it is pseudometrizable with the metric $d(x,y)=0$. \end{itemize}