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23 Jul 2014
Noetherian type in topological products
 Menachem Kojman,
 David Milovich,
 Santi Spadaro
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The cardinal invariant Noetherian type Nt(X) of a topological space X was introduced by Peregudov in 1997 to deal with base properties that were studied by the Russian School as early as 1976. We study its behavior in products and boxproducts of topological spaces.
We prove in Section 2:
(1)
There are spaces X and Y such that Nt(X×Y)< min{Nt(X), Nt(Y)}.
(2)
In several classes of compact spaces, the Noetherian type is preserved by the operations of forming a square and of passing to a dense subspace.
The Noetherian type of the Cantor Cube of weight \({\aleph _\omega }\) with the countable box topology, \({({2^{{\aleph _\omega }}})_\delta }\), is shown in Section 3 to be closely related to the combinatorics of covering collections of countable subsets of \({\aleph _\omega }\). We discuss the influence of principles like \({\square _{{\aleph _\omega }}}\) and Chang’s conjecture for \({\aleph _\omega }\) on this number and prove that it is not decidable in ZFC (relative to the consistency of ZFC with large cardinal axioms).
Within PCF theory we establish the existence of an (ℵ_{4}, ℵ_{1})sparse covering family of countable subsets of \({\aleph _\omega }\) (Theorem 3.20). From this follows an absolute upper bound of ℵ_{4} on the Noetherian type of \({({2^{{\aleph _\omega }}})_\delta }\). The proof uses a method that was introduced by Shelah in 1993 [33].
The first author was supported by a fellowship from the Institute for Advanced Study, Princeton, NJ, while working on this research.
The third author was partially supported by the Center for Advanced Studies in Mathematics at Ben Gurion University and by an INdAMCofund Outgoing fellowship. He wishes to thank the Institute for Advanced Study, Princeton, NJ and the Fields Institute of the University of Toronto for their hospitality in June 2011 and since March 2012, respectively.
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 Title
 Noetherian type in topological products
 Journal

Israel Journal of Mathematics
Volume 202, Issue 1 , pp 195225
 Cover Date
 20140701
 DOI
 10.1007/s1185601411014
 Print ISSN
 00212172
 Online ISSN
 15658511
 Publisher
 The Hebrew University Magnes Press
 Additional Links
 Topics
 Industry Sectors
 Authors

 Menachem Kojman ^{(1)}
 David Milovich ^{(2)}
 Santi Spadaro ^{(3)}
 Author Affiliations

 1. Department of Mathematics, BenGurion University of the Negev, P.O.B. 653, Be’er Sheva, 84105, Israel
 2. Department of Engineering, Mathematics and Physics, Texas A&M International University, 5201 University Blvd, Laredo, TX, 78041, USA
 3. Department of Mathematics and Statistics, Faculty of Science and Engineering, York University, Toronto, ON, M3J 1P3, Canada
