COMPACT-OPEN TOPOLOGY

In mathematics, the 'compact-open topology' is a topology defined on the set of continuous maps between two topological spaces. The compact-open topology is one of the commonly-used topologies on function spaces, and is applied in homotopy theory and functional analysis.

Contents

Definition

Properties

Definition

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'').)

Properties

★ If ''Y'' is ''T''₀, ''T''₁, Hausdorff, or regular, then the compact-open topology has the corresponding separation axiom.

★ If ''X'' is Hausdorff and ''S'' is a subbase for ''Y'', then the collection {''V''(''K'', ''U'') : ''U'' in ''S''} is a subbase for the compact-open topology on ''C''(''X'', ''Y'').

★ 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 compact convergence. In other words, if ''Y'' is a uniform space, then a sequence {''f''_''n''} converges 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'') $mapsto$ ''g''o''f'', is continuous, where all the function 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, where ''X'' is regarded as the function space ''C''(∗, ''X'') of maps from the one-point space to ''X''.

★ 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'').