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European Journal of Mathematics

, Volume 4, Issue 2, pp 664–675 | Cite as

Many Eberlein–Grothendieck spaces have no non-trivial convergent sequences

  • Vladimir V. TkachukEmail author
Research Article

Abstract

We establish that a monolithic compact space X is not scattered if and only if Open image in new window has a dense subset without non-trivial convergent sequences. Besides, for any cardinal \(\kappa \geqslant \mathfrak {c}\), the space \(\mathbb {R}^\kappa \) has a dense subspace without non-trivial convergent sequences. If X is an uncountable \(\sigma \)-compact space of countable weight, then any dense set Open image in new window has a dense subspace without non-trivial convergent sequences. We also prove that for any countably compact sequential space X, if Open image in new window has a dense k-subspace, then X is scattered.

Keywords

Function space Compact space Non-trivial convergent sequence k-space Sequential space Fréchet–Urysohn space Scattered space Eberlein–Grothendieck space EG-space Banakh property 

Mathematics Subject Classification

Primary 54C35 54C05 Secondary 54G20 

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Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  1. 1.Departamento de MatemáticasUniversidad Autónoma MetropolitanaMexico D.F.Mexico

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