Quasi-uniform convergence

A generalization of uniform convergence. A sequence of mappings $\{f_n\}$ from a topological space $X$ into a metric space $Y$ converging pointwise to a mapping $f$ is called quasi-uniformly convergent if for any $\epsilon>0$ and any positive integer $N$ there exist a countable open covering $\{\Gamma_0,\Gamma_1,\ldots\}$ of $X$ and a sequence $n_0,n_1,\ldots$ of positive integers greater than $N$ such that $\rho(f(x),f_{n_k}(x))<\epsilon$ for every $x\in\Gamma_k$. 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)