nLab
compact-open topology

Contents

Idea

A natural topology on mapping spaces of continuous functions.

Definition

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.

Properties

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.

References

  • 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? (89.15.239.172)