Dikran Dikranjan and Dmitri Shakhmatov
Let c denote the cardinality of the continuum. Using forcing we produce a model of ZFC + CH with 2^c ``arbitrarily large'' and, in this model, obtain a characterization of Abelian groups G (necessarily of size at most 2^c) which admit:
(i) a hereditarily separable group topology,
(ii) a group topology making G into an S-space,
(iii) a hereditarily separable group topology that is either precompact, or pseudocompact, or countably compact (and which can be made to contain no infinite compact subsets),
(iv) a group topology making G into an S-space that is either precompact, or pseudocompact, or countably compact (and which also can be made without infinite compact subset if necessary).As a by-product, we completely describe the algebraic structure of Abelian groups of size at most 2^c possess, at least consistently, a countably compact group topology (without infinite compact subsets, if desired).
We also put to rest a 1980 problem of van Douwen about the cofinality of the size of countably compact Abelian groups.
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Archive versions of November 14, 2002 and December 9, 2002 are still available.
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