On a supertopological cardinal invariant
type: Item
instance of: scholarly article
title: On a supertopological cardinal invariant
zbMATH Open document ID: 0562.54008
published in: Moscow University Mathematics Bulletin
publication date: 1984-01-01T00:00:00Z
review text: Tikhonov spaces X and Y are said to be t-equivalent if \(C_ p(X)\) is homeomorphic to \(C_ p(Y)\). In 1981 A. V. Arkhangel'skij proved that the Souslin number c(X) of a compact space X is equal to the supremum of weights of compact subspaces lying in \(C_ p(X)\). Thus the Souslin numbers of t-equivalent compact spaces coincide. However, it was an open problem for Tikhonov spaces. The new cardinal invariant \(p(X)=\sup \{| \gamma |:\) \(\gamma\) is a point-finite family of open non- empty subsets of \(X\}\) is introduced. It is known that \(p(X)=c(X)\) for any Čech-complete space X. The author proved that \(p(X)=a(C_ p(X))\) for an arbitrary Tikhonov space X, where \(a(Z)=sum\{| A|:\) \(A\subseteq Z\), A is homeomorphic to Alexandroff compactification of some infinite discrete set\(\}\). In particular, \(p(X)=p(Y)\) for any t- equivalent spaces X, Y and \(c(X)=c(Y)\) for any t-equivalent Čech- complete spaces X, Y. Recently the author constructed two spaces X and Y such that the free topological groups F(X) and F(Y) are isomorphic, but \(c(X)<c(Y)\). This answers Arkhangelskij's question in the negative.
Mathematics Subject Classification ID: 54A25
Mathematics Subject Classification ID: 54C35
zbMATH DE Number: 3895870
zbMATH Keywords: space of all continuous real functions
zbMATH Keywords: topology of pointwise convergence
zbMATH Keywords: supertopological cardinal invariant
zbMATH Keywords: Souslin number
zbMATH Keywords: supremum of weights of compact subspaces
zbMATH Keywords: Tikhonov spaces
zbMATH Keywords: Alexandroff compactification
zbMATH Keywords: t-equivalent Čech-complete spaces
zbMATH Keywords: free topological groups
author: Vladimir Tkachuk
reviewed by: Mikhail Tkachenko
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 3895870