# Quasi-uniform convergence

From Encyclopedia of Mathematics

A generalization of uniform convergence. A sequence of mappings from a topological space into a metric space converging pointwise to a mapping is called quasi-uniformly convergent if for any and any positive integer there exist a countable open covering of and a sequence of positive integers greater than such that for every . Uniform convergence implies quasi-uniform convergence. For sequences of continuous functions, quasi-uniform convergence is a necessary and sufficient condition for the limit function to be continuous (the Arzelà–Aleksandrov theorem).

#### References

[1] | P.S. Aleksandrov, "Einführung in die Mengenlehre und die Theorie der reellen Funktionen" , Deutsch. Verlag Wissenschaft. (1956) (Translated from Russian) |

**How to Cite This Entry:**

Quasi-uniform convergence.

*Encyclopedia of Mathematics.*URL: http://encyclopediaofmath.org/index.php?title=Quasi-uniform_convergence&oldid=14862

This article was adapted from an original article by V.V. Fedorchuk (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article