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 theoretic properties analogous to properties introduced by Menger (1924), Hurewicz (1925), Rothberger (1938), and Scheepers (1996). The studied properties were introduced independently by Ko\v{c}inac and Okunev. We obtain purely combinatorial characterizations of these properties, and combine them with other techniques to solve several questions of Babinkostova, Ko\v{c}inac, and Scheepers.Comment: To appear in IJ

Selection principles and special sets of reals

“…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.…”

“…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 .…”

Rothberger bounded groups and Ramsey theory

“…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.…”

A homeomorphism between strong measure zero sets whose graph is not of strong measure zero

“…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.…”