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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 

Notes

Acknowledgments

The authors are deeply indebted to Professor R. Pol for fruitful discussion on the Ascoli property in function spaces. We would like to thank the referee for valuable remarks and suggestions.

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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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