54-00-CompactopenTopology.tex

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\pmcreated{2013-03-22 13:25:26}
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\pmowner{antonio}{1116}
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\pmtitle{compact-open topology}
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\pmtype{Definition}
\pmcomment{trigger rebuild}
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\pmsynonym{topology of compact convergence}{CompactopenTopology}
\pmrelated{UniformConvergence}

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\begin{document}
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Let $X$ and $Y$ be topological spaces, and let $C(X,Y)$ be the set of continuous maps from $X$ to $Y.$ Given a compact subspace $K$ of $X$ and an open set $U$ in $Y,$ let 
\[
\fU_{K,U} := \set{f\in C(X,Y):\: f(x)\in U\, \text{whenever}\, x\in K}.
\]

Define the {\em compact-open topology} on $C(X,Y)$ to be the topology generated by the subbasis 
\[
\set{\fU_{K,U}:\: K\subset X\,\text{compact,}\quad U\subset Y\, \text{open} }.
\]

If $Y$ is a uniform space (for example, if $Y$ is a metric space), then this  is the topology of uniform convergence on compact sets. That is, a sequence $\seq{f_n}$ converges to $f$ in the compact-open topology if and only if for every compact subspace $K$ of $X,$ $\seq{f_n}$ converges to $f$ uniformly on $K$. If in addition $X$ is a compact space, then this is the topology of uniform convergence.
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