Let ''X'' and ''Y'' be two topological spaces, and let ''C''(''X'', ''Y'') denote the set of all continuous maps between ''X'' and ''Y''. Given a Compact Subset ''K'' of ''X'' and an Open Subset ''U'' of ''Y'', let ''V''(''K'', ''U'') denote the set of all functions ''f'' in ''C''(''X'', ''Y'') such that ''f''(''K'') is contained in ''U''. Then the collection of all such ''V''(''K'', ''U'') is a Subbase for the compact-open topology. (This collection does not always form a Base for a topology on ''C''(''X'', ''Y'').)
- If ''S'' is a for the compact-open topology.
- If ''Y'' is a Uniform Space (in particular, if ''Y'' is a Metric Space ), then the compact-open topology is equal to the topology of Uniform Convergence on Compact Set s. In other words, if ''Y'' is a uniform space, then a Sequence {''f''''n''} Converge s to ''f'' in the compact-open topology if and only if for every compact subset ''K'' of ''X'', {''f''''n''} converges uniformly to ''f'' on ''K''. In particular, if ''X'' is compact and ''Y'' is a uniform space, then the compact-open topology is equal to the topology of Uniform Convergence .
- If ''X'', ''Y'' and ''Z'' are topological spaces, and if ''X'' is a Locally Compact Regular space (not necessarily Hausdorff ), then the Composition map ''C''(''Z'', ''X'') × ''C''(''X'', ''Y'') → ''C''(''Z'', ''Y''), given by (''f'', ''g'') ''g''o''f'', is continuous, where all the functions spaces are given the compact-open topology, and where ''C''(''Z'', ''X'') × ''C''(''X'', ''Y'') is given the Product Topology . In particular, if ''X'' is a locally compact regular space, then the evaluation map ''e'' : ''X'' × ''C''(''X'', ''Y'') → ''Y'' defined by ''e''(''x'', ''f'') = ''f''(''x'') is continuous.
- If ''X'' is compact, and if ''Y'' is a metric space with metric ''d'', then the compact-open topology on ''C''(''X'', ''Y'') is Metrisable , and a metric for it is given by ''e''(''f'', ''g'') = Sup {''d''(''f''(''x''), ''g''(''x'')) : ''x'' in ''X''}, for ''f'', ''g'' in ''C''(''X'', ''Y'').
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