compact-open topology



A natural topology on mapping spaces of continuous functions.


Let XX and YY be topological spaces. The set Map(X,Y)Map(X,Y) (often denoted also C(X,Y)C(X,Y)) of continuous maps from XX to YY has a natural topology called the compact-open topology: a subbase of that topology consists of sets of the form U K,VU_{K,V}, where KXK\subset X is compact and VYV\subset Y is open?, which consists of all continuous maps f:XYf:X\to Y such that f(K)Vf(K)\subset V.

If YY is a metric space then the compact-open topology is the topology of uniform convergence on compact subsets in the sense that f nff_n \to f in Map(X,Y)Map(X,Y) with the compact-open topology iff for every compact subset KXK\subset X, f nff_n \to f uniformly on KK. If (in addition) the domain XX is compact then this is the topology of uniform convergence.


The compact-open topology is most sensible when the topology of XX is locally compact Hausdorff, for in this case Map(X,Y)Map(X,Y) with the compact-open topology is an exponential object Y XY^X in the category Top of all topological spaces. This implies the exponential law for spaces , i.e. the adjunction map is a bijection Top(X,Map(Y,Z))Top(X×Y,Z)Top(X,Map(Y,Z))\cong Top(X\times Y,Z) whenever YY is locally compact Hausdorff; and it becomes a homeomorphism Map(X,Map(Y,Z))Map(X×Y,Z)Map(X,Map(Y,Z))\cong Map(X\times Y,Z) if in addition XX is also Hausdorff. See also convenient category of topological spaces.


  • Wikipedia entry

  • Ralph H. Fox, On Topologies for Function Spaces , Bull. AMS 51 (1945) pp.429-432. (pdf)

Revised on October 18, 2014 06:39:54 by Thomas Holder? (