compact-open topology

Let X and Y be topological spacesMathworldPlanetmath, and let C(X,Y) be the set of continuous mapsMathworldPlanetmath from X to Y. Given a compactPlanetmathPlanetmath subspaceMathworldPlanetmath K of X and an open set U in Y, let


Define the compact-open topologyMathworldPlanetmath on C(X,Y) to be the topology generated by the subbasis


If Y is a uniform space (for example, if Y is a metric space), then this is the topology of uniform convergence on compact sets. That is, a sequence (fn) converges to f in the compact-open topology if and only if for every compact subspace K of X, (fn) converges to f uniformly on K. If in addition X is a compact space, then this is the topology of uniform convergence.

Title compact-open topology
Canonical name CompactopenTopology
Date of creation 2013-03-22 13:25:26
Last modified on 2013-03-22 13:25:26
Owner antonio (1116)
Last modified by antonio (1116)
Numerical id 8
Author antonio (1116)
Entry type Definition
Classification msc 54-00
Synonym topology of compact convergence
Related topic UniformConvergence