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Acta Mathematica Hungarica

, Volume 149, Issue 2, pp 324–337 | Cite as

Regular \({G_\delta}\)-diagonals and some upper bounds for cardinality of topological spaces

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Abstract

We prove that, under CH, any space with a regular \({G_{\delta}}\)-diagonal and caliber \({\omega_1}\) is separable; a corollary of this result answers, under CH, a question of Buzyakova. For any Urysohn space X, we establish the inequality \({|X|\leqq wL{(X)}^{s\Delta_2(X)\cdot{\dot(X)}}}\) which represents a generalization of a theorem of Basile, Bella, and Ridderbos. We also show that if X is a Hausdorff space, then \({|X| \leqq {(\pi\chi(X)\cdot d(X))}^{{\rm ot}(X)\cdot\psi_c(X)}}\); this result implies Šapirovskiĭ’s inequality \({|X|\leqq \pi\chi{(X)}^{c(X)\cdot\psi(X)}}\) which only holds for regular spaces. It is also proved that \({|X|\leqq \pi\chi{(X)}^{{\rm ot}(X)\cdot\psi_c(X)\cdot aL_c(X)}}\) for any Hausdorff space X; this gives one more generalization of the famous Arhangel’skii’s inequality \({|X|\le 2^{\chi(X)\cdot L(X)}}\).

Key words and phrases

cardinal function regular diagonal weakly Lindelöf number almost Lindelöf number o-tightness dense o-tightness \({\pi}\)-character \({\pi}\)-weight 

Mathematics Subject Classification

primary 54A25 secondary 54D10 54D20 

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Copyright information

© Akadémiai Kiadó, Budapest, Hungary 2016

Authors and Affiliations

  • I. S. Gotchev
    • 1
  • M. G. Tkachenko
    • 2
  • V. V. Tkachuk
    • 2
  1. 1.Department of Mathematical SciencesCentral Connecticut State UniversityNew BritainUSA
  2. 2.Departamento de MatemáticasUniversidad Autónoma MetropolitanaMexico CityMexico

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