Strong measure zero in separable metric spaces and Polish groups

Abstract

The notion of strong measure zero is studied in the context of Polish groups and general separable metric spaces. An extension of a theorem of Galvin, Mycielski and Solovay is given, whereas the theorem is shown to fail for the Baer-Specker group Z ω . The uniformity number of the ideal of strong measure zero subsets of a separable metric space is examined, providing solutions to several problems of Miller and Steprāns (Ann Pure Appl Logic 140(1-3): [52][53][54][55][56][57][58][59] 2006).

Strong measure zero sets in Polish groups

“…We shall present a proof of their theorem (the converse of Prikry's result for locally compact groups) here. Our proof follows [19].…”

“…There is a, perhaps an even more interesting, stronger ZFC conjecture on the structure of Polish groups. The following concept was introduced in [19] and the term coined in [20]: A nonempty subset C of a Polish group G is said to be anti-GMS if it is nowhere dense and for every sequence {U n : n ∈ ω} of open neighborhoods of 1 there is a sequence {g n : n ∈ ω} of elements of G such that for every g ∈ G, the set g · n∈ω g n · U n is dense in C.…”

“…The final remark of this section deals with transitive covering for category (considered by Bartoszyński [1, 2.7] for 2 ω , and Miller and Steprāns [31] for general Polish group and further studied in [19]): cov * (M(G)) = min{|A| : A ⊆ G and A · M = G for some meager set M ⊆ G} By Theorem 3.5, for a locally compact group G strong measure zero sets coincide with the sets whose meager translates do not cover G, hence, in particular, non(Smz(G)) = cov * (M(G)) for every locally compact group G. It follows from Prikry's Proposition 3.1 that cov * (M(G)) non(Smz(G)), for every Polish group G, hence cov(M) cov * (M(G)) eq for any Polish group. As non(Smz(G)) = cov(M) for all CLI groups which are not locally compact, we can conclude that: Corollary 4.…”

“…Relatively recently Kysiak [26] and Fremlin [12], independently, showed that an analogous theorem is true for all locally compact metrizable groups (see also [50]). We present a proof of Kysiak and Fremlin's result based on [19] and consider the natural question as to how far the result can be extended. The fact that the theorem does not in general hold for all Polish groups was established in [19] and [50] and extended in [20].…”

“…We present a proof of Kysiak and Fremlin's result based on [19] and consider the natural question as to how far the result can be extended. The fact that the theorem does not in general hold for all Polish groups was established in [19] and [50] and extended in [20]. This depends on further set-theoretic axioms, as the result obviously holds for all Polish groups assuming, e.g., the Borel Conjecture.…”