%%% This file is part of PlanetMath snapshot of 2009-01-12
%%% Primary Title: Tychonoff's theorem
%%% Primary Category Code: 54D30
%%% Filename: TychonoffsTheorem.tex
%%% Version: 8
%%% Owner: matte
%%% Author(s): matte, Larry Hammick, Evandar
%%% PlanetMath is released under the GNU Free Documentation License.
%%% You should have received a file called fdl.txt along with this file.        
%%% If not, please write to gnu@gnu.org.
\documentclass[12pt]{article}
\pagestyle{empty}
\setlength{\paperwidth}{8.5in}
\setlength{\paperheight}{11in}

\setlength{\topmargin}{0.00in}
\setlength{\headsep}{0.00in}
\setlength{\headheight}{0.00in}
\setlength{\evensidemargin}{0.00in}
\setlength{\oddsidemargin}{0.00in}
\setlength{\textwidth}{6.5in}
\setlength{\textheight}{9.00in}
\setlength{\voffset}{0.00in}
\setlength{\hoffset}{0.00in}
\setlength{\marginparwidth}{0.00in}
\setlength{\marginparsep}{0.00in}
\setlength{\parindent}{0.00in}
\setlength{\parskip}{0.15in}

\usepackage{html}

\usepackage{amssymb}
\usepackage{amsmath}
\usepackage{amsfonts}

\begin{document}

 Let $(X_i)_{i\in I}$ be a family of nonempty \htmladdnormallink{topological spaces}{http://planetmath.org/encyclopedia/TopologicalSpace.html}. The \htmladdnormallink{product}{http://planetmath.org/encyclopedia/Product2.html} space (see \htmladdnormallink{product topology}{http://planetmath.org/encyclopedia/Product2.html})
$$\prod_{i\in I}X_i$$
is \htmladdnormallink{compact}{http://planetmath.org/encyclopedia/Compact.html} if and only if each of the spaces $X_i$ is compact.

Not surprisingly, if $I$ is \htmladdnormallink{infinite}{http://planetmath.org/encyclopedia/InfiniteSubset.html}, the \htmladdnormallink{proof}{http://planetmath.org/encyclopedia/Proof.html} requires the \htmladdnormallink{Axiom of Choice}{http://planetmath.org/encyclopedia/MultiplicativeAxiom.html}. \htmladdnormallink{Conversely}{http://planetmath.org/encyclopedia/ConverseTheorem2.html}, one can show that Tychonoff's theorem \htmladdnormallink{implies}{http://planetmath.org/encyclopedia/VacuouslyTrue.html} that any product of nonempty sets is nonempty, which is one form of the Axiom of Choice.

\end{document}