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\author{Rebecca Gasper} \date{\today}
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\fancyhead[L]{Rebecca Gasper} 
\fancyhead[C]{Point Set Topology Table} 
\fancyhead[R]{November 20, 2008}
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\noindent\textbf{Is a compact Hausdorff space locally compact? Yes.}
\newline \textbf{Is a compact Hausdorff space locally compact Hausdorff? Yes.}
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\textbf{\underline{Claim.} All compact spaces are locally compact.}

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\begin{quote}
\textbf{Definition.} A space $X$ is said to be \textbf{locally compact at $x$} if there is some compact subspace $C$ of $X$ that contains a neighborhood of $x$.  If $X$ is locally compact at each of its points, $X$ is said simply to be \textbf{locally compact}. \hspace{.1in}-\textit{Munkres, 2nd Ed., \textsection 29}
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Since $X$ is a compact space, choose $C:=X$.  The same choice holds for each $x\in X$, giving us that $X$ is locally compact.


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