2006/09/04 by Istvan Juhasz, Juhasz, Istvan, Lajos Soukup +3
Mathematics · #54A25 (Secondary) #54A35 (Primary) 03E35 #FOS: Mathematics #General Topology (math.GN) #Logic (math.LO) #math.GN #math.LO #msc:03E35 #msc:54A25 #msc:54A35
paper · pdf · doi:10.48550/arxiv.math/0609091
arxiv created 2006/09/04 · arxiv updated 2009/12/01
In a recent paper O. Pavlov proved the following two interesting resolvability results: (1) If a space X satisfies Δ(X) > \ps(X) then X is maximally resolvable. (2) If a T3-space X satisfies Δ(X) > \pe(X) then X is ω-resolvable. Here \ps(X) (\pe(X)) denotes the smallest successor cardinal such that X has no discrete (closed discrete) subset of that size and Δ(X) is the smallest cardinality of a non-empty open set in X. In this note we improve (1) by showing that Δ(X) > \ps(X) can be relaxed to Δ(X) ≥ \ps(X). In particular, if X is a space of countable spread with Δ(X) > ω then X is maximally resolvable. The question if an analogous improvement of (2) is valid remains open, but we present a proof of (2) that is simpler than Pavlov's.