Normality in free topological groups
type: Item
instance of: scholarly article
title: Normality in free topological groups
zbMATH Open document ID: 1401.22001
DOI: 10.1134/S0001434618070283
published in: Mathematical Notes
publication date: 2018-10-30T00:00:00Z
review text: The results of the paper are related to the following open question: Is the free topological group \(F(X)\) normal if all finite powers of \(X\) are normal? The author proves the following theorem giving necessary conditions for the normality of \(F(X)\) for a countably compact space \(X\): If a space \(X\) is countably compact, then its free topological group is normal if and only if all finite powers of \(X\) are normal and countably compact. The main result is contained in the Corollary on page 276: Corollary. If an HFD exists (in particular, if CH holds), then there is a topological space \(X\) all whose finite powers are normal (moreover, the countable power \(X^\omega\) is strongly collectionwise normal and has countable tightness, and the space \(X\) itself is countably compact) such that the free topological group \(F(X)\) is not normal. The notion of an HFD-set can be found in [\textit{A. Hajnal} and \textit{I. Juhász}, Fundam. Math. 81, 147--158 (1974; Zbl 0274.54002)] and [\textit{J. Roitman}, in: Handbook of set-theoretic topology. Amsterdam: Elsevier/North Holland. 295--326 (1984; Zbl 0594.54001)].
Mathematics Subject Classification ID: 22A05
zbMATH DE Number: 6968694
zbMATH Keywords: free topological group
zbMATH Keywords: normal space
zbMATH Keywords: strongly collectionwise normal space
zbMATH Keywords: countably compact space
zbMATH Keywords: HFD-set
author: O. Pavlov
reviewed by: Mihail I. Ursul
MaRDI profile type: Publication
cites work: Q5577640
cites work: Free topological groups
cites work: Q3767521
cites work: Generalization of the Comfort-Ross theorem. I
cites work: The Product of Two Normal Initially κ-Compact Spaces
cites work: Q5795137
cites work: Free topological groups and inductive limits
cites work: Countable tightness and proper forcing
cites work: On hereditarily α-Lindelöf and α-separable spaces, II
cites work: Q3725151
cites work: Q3994935
full work available at URL: s0001434618070283
OpenAlex ID: W2895659879
Wikidata QID: Q129132176
description: scientific article; zbMATH DE number 6968694