compact-open topology
Let and be topological spaces, and let be the set of continuous maps from to Given a compact subspace of and an open set in let
Define the compact-open topology on to be the topology generated by the subbasis
If is a uniform space (for example, if is a metric space), then this is the topology of uniform convergence on compact sets. That is, a sequence converges to in the compact-open topology if and only if for every compact subspace of converges to uniformly on . If in addition is a compact space, then this is the topology of uniform convergence.
Title | compact-open topology |
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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 |