Borel resolvability of compact spaces and their subspaces

Abstract

ABSTRACT. The presence of disjoint dense (Borel) subsets in Tychonoff cubes, Borel suhspaces of Tychonoff cubes, and dyadic compacta is examined. Several problems are stated.KEY WORDS: resolvability of topological spaces, maximally, Borel, and Baire resolvable topological spaces, Borel, A-, Gs-, and F~-sets in Tychonoffcubes, A-, CA-, PCA-, and CPCA-sets in compact spaces, dyadic compactum, perfectly normal compact space, Baire space, dispersion character.

Resolvability of Lindelöf spaces

“…Malykhin (see, e.g., [1,5]) posed the following question: Let X be a Lindelöf space such that ∆(X) > ω. Is X necessarily resolvable?…”

“…Malykhin in [6] proved the ∆(X)-resolvability of finally compact topological groups of uncountable dispersion character. He also constructed an example of an irresolvable, finally compact Hausdorff space of uncountable dispersion character [5].…”

Regular spaces of small extent are ω-resolvable

“…The pairwise disjoint sets {D j : j < ω} cannot be all dense in Y because Y is not ω-resolvable, 6) and clearly Z = T ∩ D m is dense in T . Now it remains to show that…”

“…not even 2-resolvable. Since countable spaces are (hereditarily) Lindelöf, this prompted Malychin to ask the following natural question in [6]: Is every regular Lindelöf space of uncountable dispersion character (at least 2-)resolvable? He also noted that the answer to this question is negative if regular is weakened to Hausdorff.…”

Resolvability and complete accumulation points

“…In 1998 V.I. Malykhin constructed an example of irresolvable Hausdorff Lindelöf space with uncountable dispersion character and asked whether a space X is resolvable if X is regular Lindelöf and ∆(X) ≥ ω 1 [10].…”