Strong Γ-sets and Other Singular Spaces

Whereas the Gerlits-Nagy γ property is strictly weaker than the Galvin-Miller strong γ property, the corresponding strong notions for the Menger, Hurewicz, Rothberger, Gerlits-Nagy (∗), Arkhangel’skǐi and Sakai properties are equivalent to the original ones. We give new game theoretic characterizations for most of these properties, solve a related problem of Kočinac and Scheepers, and answer a question of Iliadis. 1. Conventions and summary Let X be a topological space. Throughout this paper, by open cover we mean a collection U of open subsets of X such that ∪U = X and X 6∈ U . An open cover U of X is an ω-cover of X if each finite subset of X is contained in some member of the cover. U is a γ-cover of X if each element of X belongs to all but finitely many members of U . In the celebrated paper [7], Gerlits and Nagy introduce the notion of an ǫ-space. X is an ǫ-space if each ω-cover of X contains a countable ω-cover. They prove that X is an ǫ-space if, and only if, all finite powers of X are Lindelöf. In particular, all sets of real numbers are ǫ-spaces. Another special kind of cover we will be concerned with is the finite counterpart of an ω-cover, that is, an n-cover. A cover U of X is an n-cover of X if each subset F of X of cardinality at most n is contained in some member of U . It is easy to verify the following. Lemma 1.1. Assume that U is a cover of X. U is an n-cover of X if, and only if, {U : U ∈ U} is a cover of X. Consequently, any n-cover of an ǫ-space contains a countable n-cover of that space. In light of these facts, we will confine our attention to countable open covers. This means that for some of the results, one may need to assume that the space X in question is an ǫ-space in order to obtain the analogous results for general open covers. Alternatively, if one always restricts attention to countable covers, then no assumption whatsoever is needed on the space X . Thus, unless otherwise indicated, by open cover we mean a countable collection of open subsets of X such that X 6∈ U and ∪U = X. 1991 Mathematics Subject Classification. Primary: 37F20; Secondary 26A03, 03E75 .