Selection principles and special sets of reals

Abstract

We give a selection of major open problems involving selective properties, diagonalizations, and covering properties for sets of real numbers. This is a revision of the version published as a chapter in the book Open Problems in Topology II (E. Pearl, ed.), Elsevier B.V., 2007, 91-108. The present version reports solutions of some problems, uses up-to-date notation, and has updated bibliography.Comments and further updates would be appreciated.

Products of Hurewicz Spaces in the Laver Model

“…The following theorem, which is the main result of this article, shows that an additional assumption in the results from [14,27] mentioned above is really needed. In addition, it implies that the affirmative answer to [14, Problem 2] is consistent, see [29,Section 2] for the discussion of this problem. Theorem 1.1.…”

Products of Menger spaces in the Miller model

“…First we collect some basic properties of Menger spaces. In most of the survey articles on combinatorial covering properties these are assumed to be a kind of folklore and are not stated (even without proofs) at all, see, e.g., [16,30,32,33]. The second item of the following lemma is a partial case of [31, Proposition 2.1].…”

Productively Lindelöf spaces and the covering property of Hurewicz

“…In a stream of recent papers of Tall and collaborators it was proven that under certain equalities between cardinal characteristics all metrizable productively Lindelöf spaces have strong covering properties close to the σ-compactness. In modern terminology such covering properties are called selection principles and constitute a rapidly growing area of general topology (see e.g., [22]).…”

“…If in the definition above we additionally require that {∪V n : n ∈ ω} is a γ-cover of X (this means that the set {n ∈ ω : x ∈ ∪V n } is finite for each x ∈ X), then we obtain the definition of the Hurewicz covering property introduced in [11]. Contrary to a conjecture of Hurewicz the class of metrizable spaces having the Hurewicz property appeared to be much wider than the class of σ-compact spaces [12, Theorem 5.1] (see also [6,22]).…”