Covering properties which, under weak diamond principles, constrain the extents of separable spaces

Abstract

We show that separable, locally compact spaces with property (a) necessarily have countable extent -i.e., have no uncountable closed, discrete subspaces -if the effective weak diamond principle ♦(ω, ω, <) holds. If the stronger, non-effective, diamond principle Φ(ω, ω, <) holds then separable, countably paracompact spaces also have countable extent. We also give a short proof that the latter principle implies there are no small dominating families in ω 1 ω.

Dialectical categories, cardinalities of the continuum and combinatorics of ideals

“…In a number of papers, the first author (either alone or in collaboration with Charles Morgan) has applied successfully the theory of parametrized diamond principles in order to obtain consistency and independence results (relatively to ZFC) in Set Theoretical Topology (see [20], [21], [28] and [22]).…”