No Arabic abstract
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 earlier ones, and holds in all but the most exotic models of real set theory. On the other hand, we establish productive properties f
We study products of general topological spaces with Mengers 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. Among other results, we prove that, assuming CH{}, every productively Lindelof space is productively Menger, and every productively Menger space is productively Hurewicz. None of these implications is reversible.
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 Blass--Shelah model for arbitrary values of the ultrafilter and dominating number.
W. Hurewicz proved that analytic Menger sets of reals are $sigma$-compact and that co-analytic completely Baire sets of reals are completely metrizable. It is natural to try to generalize these theorems to projective sets. This has previously been accomplished by $V = L$ for projective counterexamples, and the Axiom of Projective Determinacy for positive results. For the first problem, the first author, S. Todorcevic, and S. Tokgoz have produced a finer analysis with much weaker axioms. We produce a similar analysis for the second problem, showing the two problems are essentially equivalent. We also construct in ZFC a separable metrizable space with $omega$-th power completely Baire, yet lacking a dense completely metrizable subspace. This answers a question of Eagle and Tall in Abstract Model Theory.
Which Isbell--Mrowka spaces ($Psi$-spaces) satisfy the star version of Mengers and Hurewiczs covering properties? Following Bonanzinga and Matveev, this question is considered here from a combinatorial point of view. An example of a $Psi$-space that is (strongly) star-Menger but not star-Hurewicz is obtained. The PCF-theory function $kappamapstocof([kappa]^alephes)$ is a key tool. Using the method of forcing, a complete answer to a question of Bonanzinga and Matveev is provided. The results also apply to the mentioned covering properties in the realm of Pixley--Roy spaces, to the extent of spaces with these properties, and to the character of free abelian topological groups over hemicompact $k$ spaces.
We provide conceptual proofs of the two most fundamental theorems concerning topological games and open covers: Hurewiczs Theorem concerning the Menger game, and Pawlikowskis Theorem concerning the Rothberger game.