Compact {C}-closed spaces need not be sequential
type: Item
instance of: scholarly article
title: Compact {C}-closed spaces need not be sequential
zbMATH Open document ID: 1399.54058
author: Alan Dow
published in: Acta Mathematica Hungarica
publication date: 2018-02-13T00:00:00Z
review text: A space is C-closed, see \textit{M. Ismail} and \textit{P. Nyikos} [Topology Appl. 11, 281--292 (1980; Zbl 0434.54018)], if all of its countably compact subsets are closed. This property is implied by sequentiality and implies countable tightness in countably compact spaces. The author proves that in ``the'' Cohen model (any model obtained by adding Cohen reals to a model of the Continuum Hypothesis) compact C-closed spaces are sequentially compact and hence sequential. He then goes on to construct a model in which \(\mathbb{N}\) has a compactification \(\gamma\mathbb{N}\) with a remainder that is Fréchet-Urysohn but in which \(\mathbb{N}\) itself is completely divergent (every infinite subset has infinite subsets with disjoint closures). This space is C-closed but not sequential and answers three related questions, collected in [\textit{D. B. Shakhmatov}, in: Recent progress in general topology. Papers from the Prague Toposym 1991, held in Prague, Czechoslovakia, Aug. 19-23, 1991. Amsterdam: North-Holland. 571--640 (1992; Zbl 0801.54001)] in one fell swoop.
reviewed by: K. P. Hart
Mathematics Subject Classification ID: 54D55
Mathematics Subject Classification ID: 03E35
Mathematics Subject Classification ID: 03E75
Mathematics Subject Classification ID: 54A35
Mathematics Subject Classification ID: 54D30
Mathematics Subject Classification ID: 54D35
Mathematics Subject Classification ID: 54D80
zbMATH DE Number: 6837922
zbMATH Keywords: C-closed
zbMATH Keywords: countable tightness
zbMATH Keywords: point of first-countability
zbMATH Keywords: sequential
MaRDI profile type: Publication
full work available at URL: s10474-017-0739-x
OpenAlex ID: W2754131374
cites work: Q5396790
cites work: On spaces in which countably compact sets are closed, and hereditary properties
cites work: Minimally generated Boolean algebras
cites work: Forcing minimal extensions of Boolean algebras
cites work: Q4281194
DOI: 10.1007/S10474-017-0739-X
description: scientific article; zbMATH DE number 6837922