Compact-open topology

<p>In <a href="/facts/Mathematics/pxTouaz4">mathematics</a>, the compact-open topology is a <a href="/facts/Topological_space/v2ut7CbK">topology</a> defined on the <a href="/facts/Set_(mathematics)/BfucfHMq">set</a> of <a href="/facts/Continuous_function/sKbl02pB">continuous maps</a> between two <a href="/facts/Topological_space/v2ut7CbK">topological spaces</a>. The compact-open topology is one of the commonly used topologies on <a href="/facts/Function_space/lLUE2R1t">function spaces</a>, and is applied in <a href="/facts/Homotopy_theory/MJwYyRny">homotopy theory</a> and <a href="/facts/Functional_analysis/MIp233MT">functional analysis</a>.  It was introduced by <a href="/facts/Ralph_Fox/GFlG3J5v">Ralph Fox</a> in 1945.
</p><p>If the <a href="/facts/Codomain/MGc43nwQ">codomain</a> of the <a href="/facts/Function_(mathematics)/CdReEKCh">functions</a> under consideration has a <a href="/facts/Uniform_space/GKumHlHo">uniform structure</a> or a <a href="/facts/Metric_space/gnBBRXtc">metric structure</a> then the compact-open topology is the "topology of <a href="/facts/Uniform_convergence/ecz9eNWn">uniform convergence</a> on <a href="/facts/Compact_set/d0cgXJH7">compact sets</a>." That is to say, a <a href="/facts/Sequence/gV2S4uqp">sequence</a> of functions <a href="/facts/Limit_(mathematics)/j8JBhj4e">converges</a> in the compact-open topology precisely when it converges uniformly on every compact subset of the <a href="/facts/Domain_of_a_function/FQvoeUpX">domain</a>.
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Compact-open topology open-in-new

Compact-open topology