Topology Proceedings 28 No. 2 (2004), pp. 401-424

Tampering with pseudocompact groups

W. W. Comfort

This paper is an extended version of the Invited Address presented by the author at the 2003 Summer Conference on General Topology and Its Applications (Howard University, Washington, DC). Let C, W, CC, P and TB denote, respectively, the class of Hausdorff topological groups which are compact, w-bounded, countably compact, pseudocompact, and totally bounded, and let X, Y in { C, W, CC, P, TB }. In Part I the author attempts a survey of the literature concerning the following 5 + (2 ×5 ×5) = 55 questions and their Abelian analogues: 1. Is there an algebraic characterization of those groups G which admit a group topology T such that (G, T) Î X? 2. If (G, T) Î X, does G admit a proper dense subgroup H Î Y? 3. If (G, T) Î X does G admit a group topology S, properly refining T, such that (G, S) Î Y? Emphasis is both on what is known and on some of the most attractive or compelling unsolved questions. Part II studies more intensively the extensive literature concerning Questions 2 and 3 in the case that X and Y are the class of nonmetrizable pseudocompact (Abelian) groups. Not even a ZFC-consistent answer is known in either case, but the questions are shown to be unexpectedly related. Most of the new results cited derive from (as yet unpublished) joint work with Jorge Galindo.

Keywords: Compact group, countably compact group, w-bounded group, pseudocompact group, totally bounded group. free topological group, free Abelian topological group. refinement topology, dense subgroup. extremal pseudocompact topological group

Mathematics Subject Classification: 54H11 22A05 (22-02 54-02)

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Topology Proceedings, Volume 28, No. 2 (2004)
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