Regular spaces of small extent are ω-resolvable

Abstract

Abstract. We improve some results of Pavlov and of Filatova, respectively, concerning a problem of Malychin by showing that every regular space X that satisfies ∆(X) > e(X) is ω-resolvable. Here ∆(X), the dispersion character of X, is the smallest size of a non-empty open set in X and e(X), the extent of X, is the supremum of the sizes of all closed-and-discrete subsets of X. In particular, regular Lindelöf spaces of uncountable dispersion character are ω-resolvable.We also prove that any regular Lindelöf space X with |X| = ∆(X) = ω 1 is even ω 1 -resolvable. The question if regular Lindelöf spaces of uncountable dispersion character are maximally resolvable remains wide open.

On the resolvability of Lindelöf-generated and (countable extent)-generated spaces

“…Motivated by the fact, also due to Hewitt, that countable regular irresolvable spaces exist, Malychin asked in [10] if regular Lindelöf spaces of uncountable dispersion character are 2-resolvable. This question was answered affirmatively by Filatova in [3] and this result was strengthened in [9,Theorem 3.1] to: Theorem 1.2. Every regular space of countable extent and uncountable dispersion character is ω-resolvable.…”

“…The following result, that will play an essential role below, is a modification of the κ = ω 1 particular case of Theorem 2.7 in [9]. Theorem 1.8.…”

“…The following statement is the κ = ω 1 particular case of Lemma 2.6. (2) in [9]. Actually, in [9] it is assumed that X has countable extent but it is easy to check that this assumption is not used in the proof of part (2) of Lemma 2.6 of [9].…”

Resolvability and complete accumulation points

“…This answered Malykhin's question, but raised a natural question of 3-resolvability. In 2007 and 2012 I. Juhasz, L. Soukup and Z. Szentmiklossy have strengthened results of Pavlov and Filatova and finally proved that X is ω-resolvable [13], [14]. Also they proved that if, in addition, |X| = ∆(X) = ω 1 then X is even ω 1 -resolvable.…”

“…Also they proved that if, in addition, |X| = ∆(X) = ω 1 then X is even ω 1 -resolvable. In [14] they ask if such space X is maximally resolvable.…”