Rothberger bounded groups and Ramsey theory

Abstract

OverviewBoundedness properties in topological groups are counterparts for covering properties in general topological spaces:Guran's notion of ℵ 0 -boundedness is a counterpart of the Lindelöf covering property [9], while Okunev's notion of o-boundedness (named Menger-boundedness by Kočinac, who introduced this notion independently) and Tkachenko's corresponding property of strict o-boundedness are counterparts of σ -compactness [10]. In this paper we consider a boundedness property which approximates Borel's metric notion of strong measure zero. This boundedness property was considered in unpublished work by Galvin, was later independently considered by Kočinac under the name of Rothberger boundedness, the terminology used since [1]. It should be noted, however, that the property now called "Rothberger bounded" was already considered by Rothberger in the Hilfssatz appearing on p. 51 of [16].In [1] it was shown that a subgroup (or subset) of a metrizable topological group is Rothberger bounded if, and only if, it is strong measure zero in all left invariant metrics of the group. In [1] we also extended some of the characterizations of strong measure zero of [19] to Rothberger boundedness, but we did not have techniques to also extend the Ramseytheoretic characterization to this context. In [13] Kočinac further extends notions of boundedness in topological groups to corresponding notions of boundedness in uniform spaces. One objective of our paper is to show how to extend the Ramseytheoretic characterization of the metric notion of strong measure zero to a Ramsey-theoretic characterization of Rothberger

Weak covering properties and selection principles

“…✷ There are many ZFC examples of Lindelöf P-spaces. For example in [30] it is shown that for each uncountable cardinal number κ there is a T 3 1 2 Lindelöf P -group of cardinality κ on which TWO has a winning strategy in the game G ω 1 (Ω, Γ). Since the topology is the G δ topology, this means that in these examples TWO has a winning strategy in the game G 1 (G K , G Γ ).…”

Some remarks on topologized groups

“…In some classes of topological groups R-boundedness has been characterized Ramsey-theoretically. For example, in [19] it was done for the class of σ-compact metrizable groups.…”

Ramsey Theory and the Borel Conjecture

“…This result was proven in [36], Theorem 9, for the special context of σ-compact metric spaces. Its generalization to Theorem 3.6 was briefly explained previously in Theorem 3 of [40]. For the reader's convenience we provide a proof of Theorem 3.6 here.…”

“…For example Example 2.7. In [40] Proposition 6 it is shown that there is for each uncountable cardinal κ a T 0 topological group that has cardinality κ, does not embed as a closed subgroup into any σ-compact group, and yet TWO has a winning strategy in the game G 1 (O nbd , O) in this group. Even more, TWO has a winning strategy in the game G 1 (O, O) in this group.…”