# Topology of uniform convergence

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 .

#### References

[1] | N. Bourbaki, "General topology" , Elements of mathematics , Springer (1988) (Translated from French) |

[2] | J.L. Kelley, "General topology" , Springer (1975) |

#### Comments

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.

#### References

[a1] | R. Engelking, "General topology" , Heldermann (1989) |

**How to Cite This Entry:**

Topology of uniform convergence. V.I. Sobolev (originator),

*Encyclopedia of Mathematics.*URL: http://www.encyclopediaofmath.org/index.php?title=Topology_of_uniform_convergence&oldid=14240