On decomposition of a space into certain subsets and \({\text cl}\)- cardinality of a topological space
type: Item
instance of: scholarly article
title: On decomposition of a space into certain subsets and \({\text cl}\)- cardinality of a topological space
zbMATH Open document ID: 0806.54002
DOI: 10.1016/0166-8641(94)90056-6
author: Yu. H. Bregman
author: B. Eh. Shapirovskij
author: Alexander Šostak
published in: Topology and its Applications
publication date: 1994-08-22T00:00:00Z
review text: The idea is developed to estimate the power of a space \(X\) by means of the least cardinal number \(\kappa\) which is sufficient to decompose each subset of \(X\) into \(\kappa\) sets with certain property; see also the same authors' paper [ibid. 307-316 (1994; see the review below)]. The authors introduce the cardinal invariant \(\text{clard} (X)\) to be the least cardinal \(\kappa\) such that each subset of \(X\) is the union of a family of size at most \(\kappa\) consisting of closed subsets of \(X\). The main result of the paper says that if \(X\) is a \(T_ 1\)-space and \(w(X) \leq | X |\), then the power of \(X\) is smaller than the cofinality of \(2^{\text{clard} (X) + \ell (X)}\). In particular, under the assumption of the Generalized Luzin Hypothesis, \(| X | = \text{clard} (X) + \ell(X)\). It remains, however, the open question of whether the last equality is valid in ZFC.
Mathematics Subject Classification ID: 54A25
Mathematics Subject Classification ID: 54A35
zbMATH DE Number: 600044
zbMATH Keywords: Cl-cardinality
zbMATH Keywords: weight
zbMATH Keywords: Lindelöf number
zbMATH Keywords: weakly additional space
reviewed by: Aleksander Błaszczyk
MaRDI profile type: Publication
full work available at URL: 0166-8641(94)90056-6
OpenAlex ID: W2010402352
cites work: Q3049484
cites work: A theorem on decomposition into compact-scattered subspaces and cardinality of topological spaces
cites work: On partition of topological spaces
cites work: Q5538734
cites work: Q4150804
cites work: Q3891408
cites work: On the cardinality of certain Hausdorff spaces
cites work: Q3938195
cites work: Q5541833
cites work: Q3337413
cites work: The Existence of Q-Sets is Equivalent to the Existence of Strong Q-Sets
cites work: Q4064169
cites work: Q3345418
description: scientific article; zbMATH DE number 600044