Resolvability of spaces having small spread or extent

Topology and its Applications, 2009

Journal of Algebra, 2005

Topology and its Applications, 2006

We introduce a ZFC method that enables us to build spaces (in fact special dense subspaces of certain Cantor cubes) in which we have "full control" over all dense subsets. Using this method we are able to construct, in ZFC, for each uncountable regular cardinal λ a 0-dimensional T 2 , hence Tychonov, space which is µ-resolvable for all µ < λ but not λ-resolvable. This yields the final (negative) solution of a celebrated problem of Ceder and Pearson raised in 1967: Are ω-resolvable spaces maximally resolvable? This method enables us to solve several other open problems concerning resolvability as well.

International Journal of …, 1993

In this paper some properties of open hereditarily irresolvable spaces are obtained and the topology for a minimal irresolvable space is specified. Maximal resolvable spaces are characterized in the last section. .KEY WDRDS AND PHRASES. Resolvable, irresolvable, minimal topologies, maximal topologies. 1992 AMS SUBJECT CLASSIFICATION CODES. 54AI0. i. INTRODUCTION.

arXiv (Cornell University), 2006

We introduce a ZFC method that enables us to build spaces (in fact special dense subspaces of certain Cantor cubes) in which we have "full control" over all dense subsets. Using this method we are able to construct, in ZFC, for each uncountable regular cardinal λ a 0-dimensional T 2 , hence Tychonov, space which is µ-resolvable for all µ < λ but not λ-resolvable. This yields the final (negative) solution of a celebrated problem of Ceder and Pearson raised in 1967: Are ω-resolvable spaces maximally resolvable? This method enables us to solve several other open problems concerning resolvability as well.

2005

Fundamenta Mathematicae, 2015

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.

We continue the study of almost-ω-resolvable spaces started in [A. Tamariz-Mascarúa and H. Villegas-Rodríguez, Commentat. Math. Univ. Carol. 43, No. 4, 687–705 (2002; Zbl 1090.54011)]. We prove in ZFC: (1) every crowded T 0 space with countable tightness and every T 1 space with π-weight ≤ℵ 1 is hereditarily almost-ω-resolvable; (2) every crowded paracompact T 2 space which is the closed preimage of a crowded Fréchet T 2 space in such a way that the crowded part of each fiber is ω-resolvable, has this property too; and (3) every Baire dense-hereditarily almost-ω-resolvable space is ω-resolvable. Moreover, by using the concept of almost-ω-resolvability, we obtain two results the first one due to O. Pavlov and the other to V. I. Malykhin: (1) V=L implies that every crowded Baire space is ω-resolvable; and (2) V=L implies that the product of two crowded spaces is resolvable. Finally, we prove that the product of two almost resolvable spaces is resolvable.

Commentationes Mathematicae Universitatis Carolinae, 2016

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