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Some applications of discrete selectivity and Banakh property in function spaces

  • Vladimir V. TkachukEmail author
Research Article
  • 12 Downloads

Abstract

We establish that an uncountable space X must be essentially uncountable whenever its extent and tightness are countable. As a consequence, the equality \(\mathrm{ext}(X)= t(X)=\omega \) implies that the space \(C_{p}(X, [0,1])\) is discretely selective. If X is a metrizable space, then \(C_{p}(X, [0,1])\) has the Banakh property if and only if so does \(C_{p}(Y, [0,1])\) for some closed separable \(Y\subset X\). We apply the above results to show that, for a metrizable X, the space \(C_{p}(X, [0,1])\) is strongly dominated by a second countable space if and only if X is homeomorphic to \(D\,{\oplus }\, M\) where D is a discrete space and M is countable. For a metrizable space X, we also prove that \(C_{p}(X,[0,1])\) has the Lindelöf \(\Sigma \)-property if and only if the set of non-isolated points of X is second countable. Our results solve several open questions.

Keywords

Domination by a space Strong domination by a space Function spaces Lindelöf \(\Sigma \)-space Metrizable space Banakh property Discrete selectivity Essentially uncountable space 

Mathematics Subject Classification

54C35 54C05 46A50 

Notes

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Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Departamento de MatemáticasUniversidad Autónoma MetropolitanaMexicoMexico

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