Pseudocompact group topologies with no infinite compact subsets

2008/12/30 by Jorge Galindo, Galindo, Jorge, Sergio Macario +1
Computer Science · Mathematics · #20K99 #54H11 #Advanced Topology and Set Theory #Computability, Logic, AI Algorithms #FOS: Mathematics #General Topology (math.GN) #Group Theory (math.GR) #Rings, Modules, and Algebras

paper · doi:10.48550/arxiv.0812.5033

openalex publication_date 2008/12/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We show that every Abelian group satisfying a mild cardinal inequality admits a pseudocompact group topology from which all countable subgroups inherit the maximal totally bounded topology (we say that such a topology satisfies property \h). Every pseudocompact Abelian group G with cardinality |G|≤ 2^{2^\cc} satisfies this inequality and therefore admits a pseudocompact group topology with property \h. Under the Singular Cardinal Hypothesis (SCH) this criterion can be combined with an analysis of the algebraic structure of pseudocompact groups to prove that every pseudocompact Abelian group admits a pseudocompact group topology with property \h. We also observe that pseudocompact Abelian groups with property \h contain no infinite compact subsets and are examples of Pontryagin reflexive precompact groups that are not compact.

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