Two Classes of Metrizable Spaces $$\ell _{c}$$-Invariant

Abstract

Many properties of a Tychonoff space \(X\) have been characterized by properties of \(C_{p}(X)\) or \(C_{c}(X)\), the spaces of continuous real-valued functions on \(X\) provided with the topology of pointwise convergence or with the compact-open topology, respectively. The question of Arhangel’skii about preservation of metrizability by \(\ell _{p}\)-equivalence in the class of first countable spaces has been partially answered by Valov in the class of those first countable spaces that are Čech complete. The preservation of complete metrizability by \(\ell _{p}\)-equivalence in the class of metrizable spaces has been obtained by Baars, de Groot and Pelant. The \(\ell _{c}\)-invariance of separable metrizability and separable complete metrizability (i.e., Polish spaces) in the class of spaces of pointwise countable type has been considered very recently by K a̧ kol, López-Pellicer and Okunev. These two \(\ell _{c}\)-invariant properties were the aim of the talk given by the author in the First Meeting in Topology and Functional Analysis, dedicated to Professor Jerzy K a̧ kol on the occasion of his sixty birthday, September 27–28, in Elche (Spain). Here it is this talk with the proofs of properties needed to obtain these two \(\ell _{c}\)-invariant properties. As additional motivation for non specialist readers three classical \(C(X)\) theorems related with \(\ell \)-equivalence’s questions are also included.KeywordsČech-complete spaceCompact resolution \(C_{p}(X)\) and \(L_{p}(X)\)-spaces \(\ell _{p}\)- \(\ell _{c}\)- and \(t\)-equivalenceMetrizabilityPolish spaceSubmetrizabilitySpaces of pointwise countable type \(\mu \)-space \(\aleph _{0}\)-space2000 MSC Classification:46E1054C3546A2046A50