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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 ${{\mathbf \sharp}}$).

    Every pseudocompact Abelian group $G$ with cardinality $|G|\leq 2^{2^{\mathfrak{c}}}$ satisfies this inequality and therefore admits a pseudocompact group topology with property ${{\mathbf \sharp}}$. 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 ${{\mathbf \sharp}}$.

    We also observe that pseudocompact Abelian groups with property ${{\mathbf \sharp}}$ contain no infinite compact subsets and are examples of Pontryagin reflexive precompact groups that are not compact.
address:
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    Departmento de Matemáticas\
    Universitat Jaume I\
    Campus Riu Sec, 12071\
    Castellón Spain
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author:
- Jorge Galindo and Sergio Macario
date:
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title: Pseudocompact group topologies with no infinite compact subsets
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Introduction
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A topological space $X$ is pseudocompact if every real-valued continuous function on $X$ is bounded. Pseudocompactness is greatly enhanced by the addition of algebraic structure. This fact was discovered in 1966 by Comfort and 