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

There are spaces X and Y such that Nt(X×Y )
X), Nt(Y)}. 
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 ℵ_{ ω } with the countable box topology, \(\left( {2^{\aleph _\omega } } \right)_\delta\) , is shown in Section 3 to be closely related to the combinatorics of covering collections of countable subsets of ℵ_{ ω }. We discuss the influence of principles like □ℵ_{ ω } and Chang’s conjecture for ℵ_{ ω } 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 ℵ_{ ω } (Theorem 3.20). From this follows an absolute upper bound of ℵ_{4} on the Noetherian type of \(\left( {2^{\aleph _\omega } } \right)_\delta\) . The proof uses a method that was introduced by Shelah in 1993 [33].
 U. Abraham and M. Magidor, Cardinal arithmetic, in Handbook of Set Theory, Springer, Dordrecht, 2010, pp. 1149–1227. CrossRef
 A. V. Arhangel’skiŁ, On the metrization of topological spaces, Bulletin de l’Académie Polonaise des Sciences. Série des Sciences Mathématiques, Astronomiques et Physiques 8 (1960), 589–595.
 A. V. Arhangel’skiŁ (Ed.), General Topology III: Paracompactness, Function Spaces, Descriptive Theory, Encyclopedia of the Mathematical Sciences, Vol. 3, Springer, Berlin, 1995.
 B. Bailey, δOIF spaces, Questions and Answers in General Topology 24 (2006), 79–84.
 Z. Balogh and H. Bennett, Total paracompactness of real GOspaces, Proceedings of the American Mathematical Society 101 (1987), 753–760. CrossRef
 Z. Balogh, H. Bennett, D. Burke, D. Gruenhage, D. Lutzer and J. Mashburn, OIF spaces, Questions and Answers in General Topology 18 (2000), 129–141.
 H. Bennett and D. Lutzer, Ordered spaces with special bases, Fundamenta Mathematicae 158 (1998), 289–299.
 J. Cummings, Notes on singular cardinal combinatorics, Notre Dame Journal of Formal Logic 46 (2005), 251–282 CrossRef
 J. Cummings, M. Foreman and M. Magidor, Squares, scales, and stationary reflection, Journal of Mathematical Logic 1 (2001), 35–98. CrossRef
 D. W. Curtis, Total and absolute paracompactness, Fundamenta Mathematicae 77 (1973), 277–283.
 A. Dow, An introduction to applications of elementary submodels to topology, Topology Proceedings 13 (1988), 17–72.
 M. Foreman and M. Magidor, A very weak square principle, Journal of Symbolic Logic 62 (1997), 175–196. CrossRef
 M. Foreman, M. Magidor and S. Shelah, Martin’s maximum, saturated ideals and nonregular ultrafilters. Part 1, Annals of Mathematics 127 (1988), 1–47. CrossRef
 S. Geschke and S. Shelah, Some notes concerning the homogeneity of boolean algebras and boolean spaces, Topology and its Applications 133 (2003), 241–253. CrossRef
 M. Gitik, Prikrytype forcings, in Handbook of Set Theory, Springer, Dordrecht, 2010, pp. 1351–1447. CrossRef
 M. Gitik and M. Magidor, The singular cardinal hypothesis revisited, in Set Theory of the Continuum (Berkeley, CA, 1989), Mathematical Sciences Research Institutte Publications, Vol. 26, Springer, New York, 1992, pp. 243–279. CrossRef
 R. W. Heath, Screenability, pointwise paracompactness, and metrization of Moore spaces, Canadian Journal of Mathematics 16 (1964), 763–770. CrossRef
 I. Juhász, Cardinal Function in TopologyTen Years Later, Mathematical Centre Tracts, Vol. 123, Mathematisch Centrum, Amsterdam, 1980.
 I. Juhász, On two problems of A. V. Arhangel’skiĭ, General Topology and its Applications 2 (1972), 151–156. CrossRef
 M. Kojman, Exact upper bounds and their uses in set theory, Annals of Pure and Applied Logic 92 (1998), 267–282. CrossRef
 M. Kojman, A short proof of the PCF theorem, preprint.
 A. Lelek, Some cover properties of spaces, Fundamenta Mathematicae 64 (1969), 209–218.
 J. P. Levinski, M. Magidor and S. Shelah, Chang’s conjecture for ℵω, Israel Journal of Mathematics 69 (1990), 161–172. CrossRef
 M. Magidor and S. Shelah, When does almost free imply free? (For groups, transversals, etc.), Journal of the American Mathematical Society 7 (1994), 769–830. CrossRef
 V. I. Malykhin, On Noetherian spaces, American Mathematical Society Translations 134 (1987), 83–91.
 D. Milovich, Noetherian types of homogeneous compacta and dyadic compacta, Topology and its Applications 156 (2008), 443–464. CrossRef
 D. Milovich, Splitting families and the Noetherian type of βωω, Journal of Symbolic Logic 73 (2008), 1289–1306. CrossRef
 S. A. Peregudov, On the Noetherian type of topological spaces, Commentationes Mathematicae Universitatis Carolinae 38 (1997), 581–586.
 S. A. Peregudov and B. É. Šhapirovskiĭ, A class of compact spaces, Soviet Math. Dokl. 17 (1976), no.5, 1296–1300.
 A. Sharon and M. Viale, Some consequences of reflection on the approachability ideal, Transactions of the American Mathematical Society 362 (2010), 4201–4212. CrossRef
 S. Shelah, Cardinal Arithmetic, Oxford University Press, 1994.
 S. Shelah, More on countably compact, locally countable spaces, Israel Journal of Mathematics 62 (1988), 302–310. CrossRef
 S. Shelah, Advances in cardinal arithmetic, in Finite and Infinite Combinatorics in Sets and Logic (Banff, AB, 1991), NATO Advanced Science Institutes Series C: Mathematical and Physical Scirnces, Vol. 411, Kluwer Academic, Dordrecht, 1993, 355–383. CrossRef
 S. Shelah, Nonreflection of the bad set for Ĭθ[λ] and pcf, Acta Mathematica Hungarica, in press. Preprint sh:1008 in Shelah’s archive.
 L. Soukup, A note on Noetherian type of spaces, arXiv:1003.3189.
 R. Telgárksy, Cscattered and paracompact spaces, Fundamenta Mathematicae 73 (1971/1972), 59–74.
 S. Todorcevic, Directed sets and cofinal types, Transactions of the American Mathematical Society 290 (1985), 711–723. CrossRef
 Title
 Noetherian type in topological products
 Journal

Israel Journal of Mathematics
 DOI
 10.1007/s1185601411014
 Print ISSN
 00212172
 Online ISSN
 15658511
 Publisher
 The Hebrew University Magnes Press
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 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
