No convenient internal characterization of spaces that are productively Lindelöf is known. Perhaps the best general result known is Alster's internal characterization, under the Continuum Hypothesis, of productively Lindelöf spaces which have a basis of cardinality at most ℵ 1 . It turns out that topological spaces having Alster's property are also productively weakly Lindelöf. The weakly Lindelöf spaces form a much larger class of spaces than the Lindelöf spaces. In many instances spaces having Alster's property satisfy a seemingly stronger version of Alster's property and consequently are productively X, where X is a covering property stronger than the Lindelöf property. This paper examines the question: When is it the case that a space that is productively X is also productively Y, where X and Y are covering properties related to the Lindelöf property.Problem 2 (E.A. Michael). If X is a productively Lindelöf space, then is X ℵ0 a Lindelöf space?2010 Mathematics Subject Classification. 54B10, 54D20, 54G10, 54G12, 54G20.
We point out that in metric spaces Haver's property is not equivalent to the property introduced by Addis and Gresham. We prove that they are equal when the space has the Hurewicz property. We prove several results about the preservation of Haver's property in products. We show that if a separable metric space has the Haver property, and the nth power has the Hurewicz property, then the nth power has the Addis-Gresham property. R. Pol showed earlier that this is not the case when the Hurewicz property is replaced by the weaker Menger property. We introduce new classes of weakly infinite dimensional spaces.In [6] Haver introduced for metric space (X, d) the following property: There is for each sequence (ε n : n < ∞) of positive real numbers a corresponding sequence (V n : n < ∞) where each V n is a pairwise disjoint family of open sets, each of diameter less than ε n , such that n<∞ V n is a cover of X. When a metric space has this property we say it has the Haver property. We consider the Haver property's relation to selection principles.Let A and B be given families of collections of subsets of some set S. Then the following symbols and statements define selection principles for the pair A, B:and {T m : m < ∞} ∈ B. • S fin (A, B): For each sequence (O m : m < ∞) of elements of A there is a sequence (T m : m < ∞) with each T m a finite subset of O m , and {T m : m < ∞} ∈ B. • S c (A, B): For each sequence (O m : m < ∞) of elements of A there is a sequence (T m : m < ∞) with each T m a pairwise disjoint family refining O m , and {T m : m < ∞} ∈ B. It is clear that S 1 (A, B) implies each of S fin (A, B) and S c (A, B). Even for very standard examples of A and B no other implications hold. For example, let S be a topological space, and let O denote the collection of open covers * Tel.
We investigate game-theoretic properties of selection principles related to weaker forms of the Menger and Rothberger properties. For appropriate spaces some of these selection principles are characterized in terms of a corresponding game. We use generic extensions by Cohen reals to illustrate the necessity of some of the hypotheses in our theorems.2000 Mathematics Subject Classification. 03E35, 54A35, 54D20. Key words and phrases. Lindelöf, Menger, Rothberger, weakly, almost, infinite game. 1 As we do not need the concept of a D-space elsewhere in our paper, it is left undefined. 2 The symbol V denotes the closure of the set V . The notation O is due to Boaz Tsaban.
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