Unbounded towers and products
Autor: | Piotr Szewczak, Magdalena Włudecka |
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Rok vydání: | 2021 |
Předmět: |
Sequence
Logic 010102 general mathematics General Topology (math.GN) Structure (category theory) 0102 computer and information sciences Topological space Space (mathematics) 01 natural sciences Tower (mathematics) 26A03 54D20 03E75 03E17 Combinatorics Set (abstract data type) Cover (topology) 010201 computation theory & mathematics FOS: Mathematics 0101 mathematics Element (category theory) Mathematics - General Topology Mathematics |
Zdroj: | Annals of Pure and Applied Logic. 172:102900 |
ISSN: | 0168-0072 |
DOI: | 10.1016/j.apal.2020.102900 |
Popis: | We investigate products of sets of reals with combinatorial covering properties. A topological space satisfies $\mathsf{S}_1(\Gamma,\Gamma)$ if for each sequence of point-cofinite open covers of the space, one can pick one element from each cover and obtain a point-cofinite cover of the space. We prove that, if there is an unbounded tower, then there is a nontrivial set of reals satisfying $\mathsf{S}_1(\Gamma,\Gamma)$ in all finite powers. In contrast to earlier results, our proof does not require any additional set-theoretic assumptions. A topological space satisfies $\Omega\choose\Gamma$ (also known as Gerlits--Nagy's property $\gamma$) if every open cover of the space such that each finite subset of the space is contained in a member of the cover, contains a point-cofinite cover of the space. We investigate products of sets satisfying $\Omega\choose\Gamma$ and their relations to other classic combinatorial covering properties. We show that finite products of sets with a certain combinatorial structure satisfy $\Omega\choose\Gamma$ and give necessary and sufficient conditions when these sets are productively $\Omega\choose\Gamma$. Comment: 19 pages |
Databáze: | OpenAIRE |
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