Monatshefte für Mathematik

, Volume 180, Issue 1, pp 39–64 | Cite as

On the \(C_k\)-stable closure of the class of (separable) metrizable spaces

Article

Abstract

Denote by \(\mathbf {C}_k[\mathfrak M]\) the \(C_k\)-stable closure of the class \(\mathfrak M\) of all metrizable spaces, i.e., \(\mathbf {C}_k[\mathfrak M]\) is the smallest class of topological spaces that contains \(\mathfrak M\) and is closed under taking subspaces, homeomorphic images, countable topological sums, countable Tychonoff products, and function spaces \(C_k(X,Y)\) with Lindelöf domain in this class. We show that the class \(\mathbf {C}_k[\mathfrak M]\) coincides with the class of all topological spaces homeomorphic to subspaces of the function spaces \(C_k(X,Y)\) with a separable metrizable space X and a metrizable space Y. We say that a topological space Z is Ascoli if every compact subset of \(C_k(Z)\) is evenly continuous; by the Ascoli Theorem, each k-space is Ascoli. We prove that the class \(\mathbf {C}_k[\mathfrak M]\) properly contains the class of all Ascoli \(\aleph _0\)-spaces and is properly contained in the class of \(\mathfrak {P}\)-spaces, recently introduced by Gabriyelyan and Kąkol. Consequently, an Ascoli space Z embeds into the function space \(C_k(X,Y)\) for suitable separable metrizable spaces X and Y if and only if Z is an \(\aleph _0\)-space.

Keywords

Metric space Function space Ascoli space \(\aleph _0\)-space \(\mathfrak {P}\)-space \(C_k\)-stable closure 

Mathematics Subject Classification

46E10 54C35 54E18 

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Copyright information

© Springer-Verlag Wien 2015

Authors and Affiliations

  1. 1.Ivan Franko National UniversityL’vivUkraine
  2. 2.Jan Kochanowski UniversityKielcePoland
  3. 3.Department of MathematicsBen-Gurion University of the NegevBeer ShevaIsrael

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