Ascoli-Arzelà-theory based on continuous convergence in an (almost) non-Hausdorff setting. Partially reprinted from the journal Applied Categorical Structures 4, No. 1 (1996). (English) Zbl 0880.54015

Bartsh, René; Dencker, Peter; Poppe, Harry

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].

Reviewer: R.A.McCoy (Blacksburg)

MSC:

54C35	Function spaces in general topology
54D30	Compactness
54E15	Uniform structures and generalizations
54D50	\(k\)-spaces

Keywords:

relatively compact sets; continuous convergence; compactness on function spaces; compactness criteria of Ascoli-Arzelà-type; even continuity; equicontinuity; compact-open topology; uniform topology