It is well known that the pointwise limit of a sequence of continuous functions need not be continuous. What precisely must be added to pointwise convergence to yield continuity of the limit was given by P. S. Alexandroff in [Einführung in die Mengenlehre und die Theorie der reellen Funktionen. Berlin: VEB Deutscher Verlag der Wissenschaften (1956; Zbl 0070.04704)].
Recall that if and are metric spaces and is a sequence of functions from to , then the sequence has the Alexandroff property with respect to if for each and there exists a strictly increasing sequence of integers such that and a countable open cover of such that , we have .
Alexandroff showed that if and are metric spaces and is a sequence of functions from to that pointwise converges to , then the Alexandroff property is equivalent to continuity of . In fact this result is true without metrizability in the domain.
In the paper under review some modifications of this classical property of Alexandroff are developed for nets of continuous functions that combined with a certain convergence yield continuity of the limits of continuous functions. For instance, in Theorem 4.11, which is the last theorem of this paper, the following equivalences are proved:
Let be a Hausdorff space and be a Hausdorff uniform space. Suppose that is a bornology on with compact base, and let be a net in -convergent to . The following conditions are equivalent:
For each nonempty compact subset of , and , there exists a finite set of indices such that , for , and a neighborhood of such that , such that ;
has the classical Alexandroff property with respect to ;
is -convergent to , where is the topology corresponding to the uniformity in whose entourages basis consists of subsets of of the form
where runs over the bornology and runs over .
In this paper the author develops the extension to the uniform space setting of the theory of strong uniform continuity and strong uniform convergence, which had been developed in the setting of metric spaces in the papers [G. Beer and S. Levi, J. Math. Anal. Appl. 350, No. 2, 568–589 (2009; Zbl 1161.54003)] and [G. Beer and S. Levi, Set-Valued Var. Anal. 18, No. 3–4, 251–275 (2010; Zbl 1236.54012)].