Continuous function spaces, (db)-spaces and strongly Hewitt spaces

Abstract

We solve the long standing problem of characterizing those Tychonov spaces X for which the function space, C κ (X), of realvalued continuous functions with the compact-open topology satisfies property (db). This property is one of a spectrum of algebraic-topological properties previously known to be distinct for locally convex spaces: In increasing strength, these properties are barrelled, Bairelike, property (db), unordered Bairelike, and Baire. Strongly Hewitt spaces, recently defined and studied by Kakol andŚliwa, are characterized here using unbounded filters on X, which shows that C κ (X) is a (db)-space. We give examples of C κ (X) distinguishing property (db) within the above spectrum and from those C κ (X) with X strongly Hewitt, answer a question of Kakol, and state some further problems.  2004 Elsevier B.V. All rights reserved. MSC: primary 54C35; secondary 54E52