View source for Quasi-uniform convergence

A generalization of [[Uniform convergence|uniform convergence]]. A sequence of mappings <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076730/q0767301.png" /> from a topological space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076730/q0767302.png" /> into a metric space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076730/q0767303.png" /> converging pointwise to a mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076730/q0767304.png" /> is called quasi-uniformly convergent if for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076730/q0767305.png" /> and any positive integer <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076730/q0767306.png" /> there exist a countable open covering <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076730/q0767307.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076730/q0767308.png" /> and a sequence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076730/q0767309.png" /> of positive integers greater than <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076730/q07673010.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076730/q07673011.png" /> for every <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076730/q07673012.png" />. 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====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  P.S. Aleksandrov,   "Einführung in die Mengenlehre und die Theorie der reellen Funktionen" , Deutsch. Verlag Wissenschaft.  (1956)  (Translated from Russian)</TD></TR></table>