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Ascoli-Arzelà-theory based on continuous convergence in an (almost) non-Hausdorff setting. (English) Zbl 0880.54015
Giuli, Eraldo (ed.), Categorical topology. Proceedings of the L’Aquila conference, August 31–September 4, 1994, L’Aquila, Italy. Dordrecht: Kluwer. 221-240 (1996).
This paper gives an exposition of Ascoli-Arzelà theory in the general setting of convergence spaces using filters and the concept of continuous convergence. The main general theorem, about the set $$C(X,Y)$$ of continuous functions from topological space $$X$$ into regular space $$Y$$, is that a subset $$F$$ of $$C(X,Y)$$ is relatively compact with respect to the continuous convergence structure if and only if $$F$$ is evenly continuous and $$\{f(x):f\in F\}$$ is relatively compact in $$Y$$ for each $$x\in X$$. This is used to derive the classical Ascoli-Arzelà theorem and other well-known versions, including the characterization of compactness in $$C(X,Y)$$ with the compact-open topology in the case that $$X$$ is a k-space.
For the entire collection see [Zbl 0844.00021].

##### MSC:
 54C35 Function spaces in general topology 54D30 Compactness 54E15 Uniform structures and generalizations 54D50 $$k$$-spaces