Topology of uniform convergence

From Encyclopedia of Mathematics
Revision as of 17:07, 7 February 2011 by (talk) (Importing text file)
(diff) ← Older revision | Latest revision (diff) | Newer revision → (diff)
Jump to: navigation, search

The topology on the space of mappings from a set into a uniform space generated by the uniform structure on , the base for the entourages of which are the collections of all pairs such that for all and where runs through a base of entourages for . The convergence of a directed set to in this topology is called uniform convergence of to on . If is complete, then is complete in the topology of uniform convergence. If is a topological space and is the set of all mappings from into that are continuous, then is closed in in the topology of uniform convergence; in particular, the limit of a uniformly-convergent sequence of continuous mappings on is a continuous mapping on .


[1] N. Bourbaki, "General topology" , Elements of mathematics , Springer (1988) (Translated from French)
[2] J.L. Kelley, "General topology" , Springer (1975)


If is a metric space with the uniform structure defined by the metric, then a basis for the open sets in is formed by the sets , and one finds the notion of uniform convergence in the form it is often encountered in e.g. analysis.


[a1] R. Engelking, "General topology" , Heldermann (1989)
How to Cite This Entry:
Topology of uniform convergence. Encyclopedia of Mathematics. URL:
This article was adapted from an original article by V.I. Sobolev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article