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

We solve the long standing problem of characterizing those Tychonov spaces X for which the function space, Cκ(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 Ka̧kol and Śliwa, are characterized here using unbounded filters on X, which shows that Cκ(X) is a (db)-space. We give examples of Cκ(X) distinguishing property (db) within the above spectrum and from those Cκ(X) with X strongly Hewitt, answer a question of Ka̧kol, and state some further problems.