Selective Screenability in Topological Groups

We examine the selective screenability property in topological groups. In the metrizable case we also give characterizations of Sc(Onbd,O) and Smirnov-Sc(Onbd,O) in terms of the Haver property and finitary Haver property respectively relative to left-invariant metrics. We prove theorems stating conditions under which Sc(Onbd,O) is preserved by products. Among metrizable groups we characterize the countable dimensional ones by a natural game. 1. Definitions and notation Let G be topological space. We shall use the notations: • O: The collection of open covers of G. An open cover U of a topological space G is said to be • an ω-cover if G is not a member of U , but for each finite subset F of G there is a U ∈ U such that F ⊂ U . The symbol Ω denotes the collection of ω covers of G. • groupable if there is a partition U = ∪n<∞Un, where each Un is finite, and for each x ∈ G the set {n : x 6∈ ∪Un} is finite. The symbol Ogp denotes the collection of groupable open covers of the space. • large if each element of the space is contained in infinitely many elements of the cover. The symbol Λ denotes the collection of large covers of the space. • c-groupable if there is a partition U = ∪n<∞Un, where each Un is pairwise disjoint and each x is in all but finitely many ∪Un. The symbol Ocgp denotes the collection of c-groupable open covers of the space. Now let (G, ∗) be a topological group with identity element e. We will assume that G is not compact. For A and B subsets of G, A ∗ B denotes {a ∗ b : a ∈ A, b ∈ B}. We use the notation A2 to denote A ∗ A, and for n > 1, An denotes An−1 ∗ A. For a neighborhood U of e, and for a finite subset F of G the set F ∗ U is a neighborhood of the finite set F . Thus,