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

T. Banakh, S. Gabriyelyan

Research output: Contribution to journalArticlepeer-review

25 Scopus citations

Abstract

Denote by Ck[ M] the Ck-stable closure of the class M of all metrizable spaces, i.e., Ck[ M] is the smallest class of topological spaces that contains M and is closed under taking subspaces, homeomorphic images, countable topological sums, countable Tychonoff products, and function spaces Ck(X, Y) with Lindelöf domain in this class. We show that the class Ck[ M] coincides with the class of all topological spaces homeomorphic to subspaces of the function spaces Ck(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 Ck(Z) is evenly continuous; by the Ascoli Theorem, each k-space is Ascoli. We prove that the class Ck[ M] properly contains the class of all Ascoli ℵ0-spaces and is properly contained in the class of P-spaces, recently introduced by Gabriyelyan and Kąkol. Consequently, an Ascoli space Z embeds into the function space Ck(X, Y) for suitable separable metrizable spaces X and Y if and only if Z is an ℵ0-space.

Original languageEnglish
Pages (from-to)39-64
Number of pages26
JournalMonatshefte fur Mathematik
Volume180
Issue number1
DOIs
StatePublished - 1 May 2016

Keywords

  • Ascoli space
  • C-stable closure
  • Function space
  • Metric space
  • P-space
  • ℵ-space

ASJC Scopus subject areas

  • Mathematics (all)

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