Spaces making continuous convergence and locally uniform convergence coincide, their very weak P-property, and their topological behaviour. (English) Zbl 0732.54005

Given a convergence (or topological) space X, continuous convergence and locally uniform convergence are two types of convergence (not necessarily topological) in the set C(X) of all real-valued, continuous functions on X; these two convergences lie between compact and uniform convergence. A convergence (or topological) space X for which continuous and locally uniform convergence are equal is called a “c\(=lu\)-space”. A characterization of such spaces in terms of “covers” was given in an earlier paper by the second author. The present paper examines the \(c=lu\) property and some other closely related properties, focusing on their behavior relative to subspaces, products, box products, various types of quotient, and topological modification.
The stated goal is to find out what \(c=lu\)-spaces really “look like”. Despite some technical terminology requiring a “short course in convergence”, the authors are reasonably successful in achieving their goal, thanks in part to their “reader-friendly” style of writing.
Reviewer: D.C.Kent (Pullman)


54A20 Convergence in general topology (sequences, filters, limits, convergence spaces, nets, etc.)
54C30 Real-valued functions in general topology
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