Selective sequential pseudocompactness

DOI10.1016/J.TOPOL.2017.02.016zbMATH Open1367.54015arXiv1702.02055OpenAlexW2594838037MaRDI QIDQ524325

Alejandro Dorantes-Aldama, Dmitri Shakhmatov

Publication date: 2 May 2017

Published in: Topology and its Applications (Search for Journal in Brave)

Abstract: We say that a topological space X is selectively sequentially pseudocompact (SSP for short) if for every sequence (U_n) of non-empty open subsets of X, one can choose a point x_n in U_n for every n in such a way that the sequence (x_n) has a convergent subsequence. We show that the class of SSP spaces is closed under taking arbitrary products and continuous images, contains the class of all dyadic spaces and forms a proper subclass of the class of strongly pseudocompact spaces introduced recently by Garc'ia-Ferreira and Ortiz-Castillo. We investigate basic properties of this new class and its relations with known compactness properties. We prove that every omega-bounded (=the closure of which countable set is compact) group is SSP, while compact spaces need not be SSP. Finally, we construct SSP group topologies on both the free group and the free Abelian group with continuum-many generators.

Full work available at URL: https://arxiv.org/abs/1702.02055

zbMATH Keywords

topological groupconvergent sequencevariety of groupsfree abelian grouppseudocompactsequentially compactstrongly pseudocompactsequentially pseudocompact

Mathematics Subject Classification ID

Compactness (54D30) Topological groups (topological aspects) (54H11) Group rings of finite groups and their modules (group-theoretic aspects) (20C05) Convergence in general topology (sequences, filters, limits, convergence spaces, nets, etc.) (54A20) Axiomatics and elementary properties of groups (20A05)

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