An example of \(t^*_p\)-equivalent spaces which are not \(t_p\)-equivalent
type: Item
instance of: scholarly article
title: An example of \(t^*_p\)-equivalent spaces which are not \(t_p\)-equivalent
zbMATH Open document ID: 0918.54013
DOI: 10.1016/S0166-8641(97)00154-5
author: Witold Marciszewski
author: Jan van Mill
published in: Topology and its Applications
publication date: 1999-08-17T00:00:00Z
review text: Recently, \textit{T. Banakh} and \textit{R. Cauty} [Colloq. Math. 73, No. 1, 25-33 (1997; Zbl 0882.57014)] proved that if \(X\) is countable and nondiscrete then \(C_p^*(X)\) is homeomorphic to \(C_p(X)\times \sigma\), where \(\sigma\) denotes the linear span of the standard basis in \(\ell^2\). This interesting result has several nontrivial consequences, among them the statement that if \(C_p(X)\) and \(C_p(Y)\) are homeomorphic then so are \(C_p^*(X)\) and \(C_p^*(Y)\). This result suggests the natural question of whether the reverse implication holds. The aim of this note is to answer this question in the negative: there exist countable spaces \(X\) and \(Y\) for which \(C_p^*(X)\approx C_p^*(Y)\) (i.e., \(X\) and \(Y\) are \(t_p^*\)-equivalent) but \(C_p(X) \not\approx C_p(Y)\) (i.e., \(X\) and \(Y\) are not \(t_p\)-equivalent). For a related result, see [\textit{J. Baars, J. de Groot, J. van Mill}, and \textit{J. Pelant}, Proc. Am. Math. Soc. 119, No. 3, 963-969 (1993; Zbl 0787.54018)].
Mathematics Subject Classification ID: 54C35
zbMATH DE Number: 1308008
zbMATH Keywords: ultrafilter
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1308008