An example of \(t^*_p\)-equivalent spaces which are not \(t_p\)-equivalent
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)].
scientific article; zbMATH DE number 1308008