2008
DOI: 10.1007/s11856-008-1060-8
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The combinatorics of the Baer-Specker group

Abstract: Denote the integers by Z and the positive integers by N. The groups Z^k (k a natural number) are discrete, and the classification up to isomorphism of their (topological) subgroups is trivial. But already for the countably infinite power Z^N of Z, the situation is different. Here the product topology is nontrivial, and the subgroups of Z^N make a rich source of examples of non-isomorphic topological groups. Z^N is the Baer-Specker group. We study subgroups of the Baer-Specker group which possess group theo… Show more

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Cited by 11 publications
(12 citation statements)
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“…Some of them are surveyed in [23]. The relations among these boundedness properties and their topological counterparts were studied in many papers, see [25,26,35,65,5,3,1,81,36], and references therein. In particular, the following diagram of implications is complete.…”
Section: γXmentioning
confidence: 99%
See 1 more Smart Citation
“…Some of them are surveyed in [23]. The relations among these boundedness properties and their topological counterparts were studied in many papers, see [25,26,35,65,5,3,1,81,36], and references therein. In particular, the following diagram of implications is complete.…”
Section: γXmentioning
confidence: 99%
“…Is there an analytic group satisfying S fin (O nbd , O) but not S 1 (Ω nbd , Γ)? It seems that Z N for boundedness properties of topological groups is like R for topological and measure theoretic notions of smallness [36]. Thus, unless otherwise indicated, all of the problems in the remainder of this section are concerning subgroups of Z N .…”
Section: γXmentioning
confidence: 99%
“…More recently in Theorem 8.5 of [14] it is shown that it is consistent, relative to the consistency of ZFC, that there are metrizable groups which are Rothberger bounded but are not Rothberger spaces.…”
Section: Some Terminology and Notationmentioning
confidence: 99%
“…T. Weiss [13] showed, answering questions from [7], that in some models of ZFC one can define subgroups G, H of Z N such that (i) G and H are strong measure zero sets with respect to the standard "first difference" metric in Z N and no σ -compact set in Z N contains G or H , (ii) G fails the Menger property, (iii) H has the Menger property, but fails the Rothberger property.…”
Section: Some Singular Subgroups Of Z Z Z N N Nmentioning
confidence: 99%