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

摘要

We solve the long standing problem of characterizing those Tychonov spaces X for which the function space, C_κ(X), of real valued 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 Bake. Strongly Hewitt spaces, recently defined and studied by Kakol and Sliwa, 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.