相关论文: Products and selection principles
We construct Luzin-type subsets of the real line in all finite powers Rothberger, with a non-Menger product. To this end, we use a purely combinatorial approach which allows to weaken assumptions used earlier to construct sets with…
We construct Menger subsets of the real line whose product is not Menger in the plane. In contrast to earlier constructions, our approach is purely combinatorial. The set theoretic hypothesis used in our construction is far milder than…
We characterize various Menger/Rothberger related properties by means of ultrafilter convergence, and discuss their behavior with respect to products.
In this paper we shall introduce some nonmeasurable and completely nonmeasurable subsets of the plane with various additional properties, e.g. being Hamel basis, intersecting each line in a strong Luzin / Sierpi\'nski set. Also some…
We study the preservation of selective covering properties, including classic ones introduced by Menger, Hurewicz, Rothberger, Gerlits and Nagy, and others, under products with some major families of concentrated sets of reals. Our methods…
We consider products of sets of reals with a combinatorial structure based on scales parameterized by filters. This kind of sets were intensively investigated in products of spaces with combinatorial covering properties as Hurewicz,…
We study productive properties of gamma spaces, and their relation to other, classic and modern, selective covering properties. Among other things, we prove the following results: 1. Solving a problem of F. Jordan, we show that for every…
In this paper we consider a notion of $\mathcal{I}$-Luzin set which generalizes the classical notion of Luzin set and Sierpi{\'n}ski set on Euclidean spaces. We show that there is a translation invariant $\sigma$-ideal $\mathcal{I}$ with…
We study products of general topological spaces with Menger's covering property, and its refinements based on filters and semifilters. To this end, we extend the projection method from the classic real line topology to the Michael topology.…
We construct, using mild combinatorial hypotheses, a real Menger set that is not Scheepers, and two real sets that are Menger in all finite powers, with a non-Menger product. By a forcing-theoretic argument, we show that the same holds in…
The aim of this note is to announce some results about the probabilistic and deterministic asymptotic properties of linear groups. The first one is the analogue, for norms of random matrix products, of the classical theorem of Cramer on…
We present a comprehensive report on the relationships between variations of the Menger and Rothberger selection properties with respect to $\omega$-covers and $k$-covers in the most general topological setting and address the finite…
We develop the semifilter approach to the classical Menger and Hurewicz covering properties and show that the small cardinal g is a lower bound of the additivity number of the family of Menger subspaces of the Baire space, and under u< g…
Tensor products of ultrafilters have special combinatorial features closely related to Ramsey's Theorem, making them useful tools in applications. Here we first review their fundamental properties and isolate some new ones, including a…
Menger conjectured that subsets of R with the Menger property must be ${\sigma}$-compact. While this is false when there is no restriction on the subsets of R, for projective subsets it is known to follow from the Axiom of Projective…
We examine the selective screenability property in topological groups. In the metrizable case we also give characterizations in terms of the Haver property and finitary Haver property respectively relative to left-invariant metrics. We…
We give conditions under which a product of topological spaces satisfies some local property. The conditions are necessary and sufficient when the corresponding global property is preserved under finite products. Further examples include…
We consider the question of when sets definable in first-order expansions of groups contain the product of two infinite sets (we refer to this as the "productset property"). We first show that the productset property holds for any definable…
Arhangelskii's properties $\alpha_2$ and $\alpha_4$ defined for convergent sequences may be characterized in terms of Scheeper's selection principles. We generalize these results to hold for more general collections and consider these…
Motivated by the model theory of higher order logics, a certain kind of topological spaces had been introduced on ultraproducts. These spaces are called ultratopologies. Ultratopologies provide a natural extra topological structure for…