sequentially compactSequentiallyCompact2002-07-06 22:50:292009-01-21 17:54:53Definitiontopologysequenceconvergence% 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 hereA topological space $X$ is \emph{sequentially compact} if every sequence in $X$ has a convergent subsequence.
For example, in a compact topological space, every net has a convergent subnet, and is therefore sequentially compact, as every subnet of a sequence is a subsequence.
When $X$ is a metric space, the following are equivalent:
\begin{itemize}
\item
$X$ is sequentially compact.
\item
$X$ is limit point compact.
\item
$X$ is compact.
\item
$X$ is totally bounded and complete.
\end{itemize}