Weakly metrizable pseudocompact groups

Dikran Dikranjan, Anna Giordano Bruno, Chiara Milan


We study various weaker versions of metrizability for pseudocompact abelian groups G: singularity (G possesses a compact metrizable subgroup of the form mG, m > 0), almost connectedness (G is metrizable modulo the connected component) and various versions of extremality in the sense of Comfort and co-authors (s-extremal, if G has no proper dense pseudocompact subgroups, r-extremal, if G admits no proper pseudocompact refinement). We introduce also weaklyextremal pseudocompact groups (weakening simultaneously s-extremal and r-extremal). It turns out that this “symmetric” version of ex-tremality has nice properties that restore the symmetry, to a certain extent, in the theory of extremal pseudocompact groups obtaining simpler uniform proofs of most of the known results. We characterize doubly extremal pseudocompact groups within the class of s-extremal pseudocompact groups. We give also a criterion for r-extremality for connected pseudocompact groups.


Pseudocompact group; Gδ-dense subgroup; Extremal pseudocompact group; Dense graph

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1. Dense minimal pseudocompact subgroups of compact Abelian groups
Anna Giordano Bruno
Topology and its Applications  vol: 155  issue: 17-18  first page: 1919  year: 2008  
doi: 10.1016/j.topol.2007.04.020

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Universitat Politècnica de València

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