Calibers and point-finite cellularity of the space Cp(X) and some questions of S. Gul'ko and M. Hušek
type: Item
instance of: scholarly article
title: Calibers and point-finite cellularity of the space Cp(X) and some questions of S. Gul'ko and M. Hušek
zbMATH Open document ID: 0591.54023
DOI: 10.1016/0166-8641(86)90017-9
published in: Topology and its Applications
publication date: 1986-01-01T00:00:00Z
review text: For any Tichonoff space X let Cp(X) denote the space of all continuous real-valued functions defined on X with the topology of pointwise convergence. A cardinal \(\tau\) is said to be a caliber of X in case every family with cardinality \(\tau\) of non-empty open subsets of X has a subfamily of cardinality \(\tau\) with a non-empty intersection. The aim of the present paper is to find conditions on a compact space X which will ensure that \(\aleph_ 1\) is a caliber of Cp(X). The authors show, among other things, that the following result is consistent with ZFC: a compact space X is metrizable if, and only if, \(\aleph_ 1\) is a caliber of Cp(X). They go on to show that a compact space X is metrizable if, and only if, \(\aleph_ 1\) and \(\aleph_ 2\) are both calibers of Cp(X), provided we assume that \(2^{\aleph_ 1}=\aleph_ 2\). For any set A let \(\Sigma_ A\) denote the set of real-valued functions f defined on A with the property that \(\{\) \(a\in A:\) \(| f(a)| \geq \epsilon \}\) is finite for every \(\epsilon >0\). Another of the author's results says that an extremally disconnected space X can be continuously and linearly injected into \(\Sigma_ A\) for some set A if, and only if, X is separable.
Mathematics Subject Classification ID: 54E35
Mathematics Subject Classification ID: 54A25
Mathematics Subject Classification ID: 54C35
Mathematics Subject Classification ID: 54D60
Mathematics Subject Classification ID: 54G05
Mathematics Subject Classification ID: 54E45
Mathematics Subject Classification ID: 54A35
zbMATH DE Number: 3949285
zbMATH Keywords: point-finite cellularity
zbMATH Keywords: tightness
zbMATH Keywords: precaliber
zbMATH Keywords: supersequence
zbMATH Keywords: Alexandroff number
zbMATH Keywords: \(\tau \) -accessibility
zbMATH Keywords: Tikhonov spaces
zbMATH Keywords: continuum
zbMATH Keywords: hypothesis
zbMATH Keywords: inaccessible set
zbMATH Keywords: topology of pointwise convergence
zbMATH Keywords: caliber
zbMATH Keywords: extremally disconnected space
author: A. V. Arhangel'skii
author: Vladimir Tkachuk
MaRDI profile type: Publication
full work available at URL: 0166-8641(86)90017-9
OpenAlex ID: W2003202704
cites work: Q3312892
cites work: Q3049484
cites work: Q4150804
cites work: Q3854164
cites work: On the small diagonals
cites work: Q5723435
cites work: Q4151370
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description: scientific article; zbMATH DE number 3949285