ProductsOfConnectedSpaces 2003-09-05 03:25:13 2004-02-18 23:12:07 Theorem connected 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} % 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 \newtheorem{Theorem}{Theorem} \PMlinkescapeword{implication} \begin{Theorem}\cite{lang, mukherjea} %{\bf Theorem} \cite{lang, mukherjea} Let $(X_i)_{i\in I}$ be a family of topological spaces. Then the product space \[\prod_{i\in I}X_i\] with the product topology is connected if and only if each space $X_i$ is connected. \end{Theorem} As is true of most results in topology involving products, the forward implication requires the axiom of choice. \begin{thebibliography}{9} \bibitem{lang} S. Lang, \emph{Analysis II}, Addison-Wesley Publishing Company Inc., 1969. \bibitem{mukherjea} A. Mukherjea, K. Pothoven, \emph{Real and Functional Analysis}, Plenum Press, 1978. \end{thebibliography}