Scales, products and the second row of the Scheepers diagram
Michał Pawlikowski, Piotr Szewczak, Lyubomyr Zdomskyy
TL;DR
Problem: determine productivity of $\mathsf{S}_1(\Gamma,\Omega)$ in products of scale-structured sets within the Scheepers diagram, including $\mathfrak{b}$-scale and $\mathfrak{d}$-concentrated constructions. Approach: develop $\kappa$-fin-unbounded scale theory, introduce the Michael topology, and leverage uniformization and Borel-cover techniques to prove product theorems; also construct counterexamples and analyze Miller-model behavior with ultrafilters. Contributions: (i) a main product theorem for $(X \cup \mathrm{Fin})^{k}_{\mathrm{M}} \times Y$ yielding $\mathsf{S}_1(\Gamma,\Omega)$, (ii) applications to finite powers of scale sets and corollaries for standard products, (iii) counterexamples showing limitations of the method, (iv) Miller-model results establishing $\mathsf{S}_1(\Gamma,\Omega)$ for $\mathfrak{d}$-concentrated sets, and (v) characterizations and open problems clarifying the structure of second-row properties. Significance: advances understanding of the second row in the Scheepers diagram for product spaces, clarifies the role of filters, ultrafilters, and forcing models in topological combinatorics, and provides robust techniques for handling $\gamma$- and $\omega$-covers in complex constructions.
Abstract
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, Scheepers, Menger and Rothberger. We will complete this picture with focusing on properties from the second row of the Scheepers diagram. In particular we show that in the Miller model a product space of two $\mathfrak{d}$-concentrated sets has a strong covering property $\mathsf{S}_1(Γ,Ω)$. We provide also counterexamples in products to demonstrate limitations of used methods.