compact-open topologyCompactOpenTopology2003-02-05 15:58:422003-02-07 00:18:33Definition% this is the default PlanetMath preamble. as your knowledge
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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.