On resolvability of topological spaces
type: Item
instance of: scholarly article
title: On resolvability of topological spaces
zbMATH Open document ID: 1012.54004
DOI: 10.1016/S0166-8641(02)00004-4
published in: Topology and its Applications
publication date: 2003-04-02T00:00:00Z
review text: A space \(X\) is \(\kappa\)-resolvable if it can be represented as the union of \(\kappa\)-many mutually disjoint dense subsets. It is maximally resolvable if it is \(\Delta(X)\)-resolvable, where \(\Delta(X)\) is the dispersion character of \(X\). The pseudospread, \(\text{ps}(X)\), (respectively, pseudoextent, \(\text{pext}(X)\)) of a space \(X\) is the minimum cardinal \(\tau^{+}\) such that \(X\) contains no discrete (respectively, closed discrete) subset of cardinality \(\tau\). This paper considers resolvability of spaces whose pseudoextent or pseudospread is less than their dispersion character. The main theorems are as follows: 1) If \(\Delta(X)> \text{ps}(X)\) then \(X\) is maximally resolvable; 2) If \(X\) has a \(\pi\)-network of regular closed sets and \(\Delta(X)> \text{pext}(X)\), then \(X\) is \(\omega\)-resolvable; 3) Under \(V=L\), a dense-in-itself Baire space is \(\omega\)-resolvable. Interesting and immediate corollaries to the first two results are that assuming the negation of the Continuum Hypothesis, a hereditarily Lindelöf connected Hausdorff space is maximally resolvable and a connected regular Lindelöf space is \(\omega\)-resolvable.
Mathematics Subject Classification ID: 54A25
Mathematics Subject Classification ID: 54A35
Mathematics Subject Classification ID: 54G20
Mathematics Subject Classification ID: 54E52
zbMATH DE Number: 1891198
zbMATH Keywords: resolvable space
zbMATH Keywords: hereditarily irresolvable space
zbMATH Keywords: maximal space
zbMATH Keywords: submaximal space
zbMATH Keywords: SI-space
zbMATH Keywords: MI-space
zbMATH Keywords: maximally resolvable space
zbMATH Keywords: pseudoextent
zbMATH Keywords: \(\tau\)-resolvable space
author: O. Pavlov
reviewed by: Richard G. Wilson
MaRDI profile type: Publication
cites work: Q4164522
cites work: On submaximal spaces
cites work: On maximally resolvable spaces
cites work: Resolvability: A selective survey and some new results
cites work: Q5621765
cites work: Regular maximal spaces
cites work: Finite and $\omega $-resolvability
cites work: Q5511988
cites work: A problem of set-theoretic topology
cites work: Q5788022
cites work: Q4720386
cites work: Q4086408
cites work: Resolvability of \(A\)-, \(CA\)- and \(PCA\)-sets in compacta
cites work: An example of a connected irresolvable Hausdorff space
cites work: Q3311274
cites work: Resolution property in generalized k-spaces
full work available at URL: s0166-8641(02)00004-4
OpenAlex ID: W2071786008
description: scientific article; zbMATH DE number 1891198