Quasi-uniform convergence

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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).


[1] P.S. Aleksandrov, "Einführung in die Mengenlehre und die Theorie der reellen Funktionen" , Deutsch. Verlag Wissenschaft. (1956) (Translated from Russian)
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Quasi-uniform convergence. Encyclopedia of Mathematics. URL:
This article was adapted from an original article by V.V. Fedorchuk (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article