# Noetherian type in topological products

## Abstract

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 box-products of topological spaces.

- (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].

## Preview

Unable to display preview. Download preview PDF.

### References

- [1]U. Abraham and M. Magidor,
*Cardinal arithmetic*, in*Handbook of Set Theory*, Springer, Dordrecht, 2010, pp. 1149–1227.CrossRefGoogle Scholar - [2]A. V. Arhangel’skiĭ,
*On the metrization of topological spaces*, Bulletin de l’Académie Polonaise des Sciences. Série des Sciences Mathématiques, Astronomiques et Physiques**8**(1960), 589–595.Google Scholar - [3]A. V. Arhangel’skiĭ (Ed.),
*General Topology III: Paracompactness, Function Spaces, Descriptive Theory*, Encyclopedia of the Mathematical Sciences, Vol. 3, Springer, Berlin, 1995.MATHGoogle Scholar - [4]B. Bailey,
*δ-OIF spaces*, Questions and Answers in General Topology**24**(2006), 79–84.MathSciNetMATHGoogle Scholar - [5]Z. Balogh and H. Bennett,
*Total paracompactness of real GO-spaces*, Proceedings of the American Mathematical Society**101**(1987), 753–760.MathSciNetCrossRefMATHGoogle Scholar - [6]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.MathSciNetMATHGoogle Scholar - [7]H. Bennett and D. Lutzer,
*Ordered spaces with special bases*, Fundamenta Mathematicae**158**(1998), 289–299.MathSciNetMATHGoogle Scholar - [8]J. Cummings,
*Notes on singular cardinal combinatorics*, Notre Dame Journal of Formal Logic**46**(2005), 251–282.MathSciNetCrossRefMATHGoogle Scholar - [9]J. Cummings, M. Foreman and M. Magidor,
*Squares, scales, and stationary reflection*, Journal of Mathematical Logic**1**(2001), 35–98.MathSciNetCrossRefMATHGoogle Scholar - [10]D. W. Curtis,
*Total and absolute paracompactness*, Fundamenta Mathematicae**77**(1973), 277–283.MathSciNetMATHGoogle Scholar - [11]A. Dow,
*An introduction to applications of elementary submodels to topology*, Topology Proceedings**13**(1988), 17–72.MathSciNetMATHGoogle Scholar - [12]M. Foreman and M. Magidor,
*A very weak square principle*, Journal of Symbolic Logic**62**(1997), 175–196.MathSciNetCrossRefMATHGoogle Scholar - [13]M. Foreman, M. Magidor and S. Shelah,
*Martin’s maximum, saturated ideals and nonregular ultrafilters. Part 1*, Annals of Mathematics**127**(1988), 1–47.MathSciNetCrossRefMATHGoogle Scholar - [14]S. Geschke and S. Shelah,
*Some notes concerning the homogeneity of boolean algebras and boolean spaces*, Topology and its Applications**133**(2003), 241–253.MathSciNetCrossRefMATHGoogle Scholar - [15]M. Gitik,
*Prikry-type forcings*, in*Handbook of Set Theory*, Springer, Dordrecht, 2010, pp. 1351–1447.CrossRefGoogle Scholar - [16]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.CrossRefGoogle Scholar - [17]R. W. Heath,
*Screenability, pointwise paracompactness, and metrization of Moore spaces*, Canadian Journal of Mathematics**16**(1964), 763–770.MathSciNetCrossRefMATHGoogle Scholar - [18]I. Juhász,
*Cardinal Function in Topology — Ten Years Later*, Mathematical Centre Tracts, Vol. 123, Mathematisch Centrum, Amsterdam, 1980.Google Scholar - [19]I. Juhász,
*On two problems of A. V. Arhangel’skiĭ*, General Topology and its Applications**2**(1972), 151–156.CrossRefMATHGoogle Scholar - [20]M. Kojman,
*Exact upper bounds and their uses in set theory*, Annals of Pure and Applied Logic**92**(1998), 267–282.MathSciNetCrossRefMATHGoogle Scholar - [21]M. Kojman,
*A short proof of the PCF theorem*, preprint.Google Scholar - [22]A. Lelek,
*Some cover properties of spaces*, Fundamenta Mathematicae**64**(1969), 209–218.MathSciNetMATHGoogle Scholar - [23]J. P. Levinski, M. Magidor and S. Shelah,
*Chang’s conjecture for*\({\aleph _\omega }\), Israel Journal of Mathematics**69**(1990), 161–172.MathSciNetCrossRefMATHGoogle Scholar - [24]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.MathSciNetCrossRefMATHGoogle Scholar - [25]V. I. Malykhin,
*On Noetherian spaces*, American Mathematical Society Translations**134**(1987), 83–91.MATHGoogle Scholar - [26]D. Milovich,
*Noetherian types of homogeneous compacta and dyadic compacta*, Topology and its Applications**156**(2008), 443–464.MathSciNetCrossRefMATHGoogle Scholar - [27]D. Milovich,
*Splitting families and the Noetherian type of βω \ ω*, Journal of Symbolic Logic**73**(2008), 1289–1306.MathSciNetCrossRefMATHGoogle Scholar - [28]S. A. Peregudov,
*On the Noetherian type of topological spaces*, Commentationes Mathematicae Universitatis Carolinae**38**(1997), 581–586.MathSciNetMATHGoogle Scholar - [29]S. A. Peregudov and B. É. Šhapirovskiĭ,
*A class of compact spaces*, Soviet Math. Dokl.**17**(1976), no. 5, 1296–1300.MATHGoogle Scholar - [30]A. Sharon and M. Viale,
*Some consequences of reflection on the approachability ideal*, Transactions of the American Mathematical Society**362**(2010), 4201–4212.MathSciNetCrossRefMATHGoogle Scholar - [31]S. Shelah,
*Cardinal Arithmetic*, Oxford University Press, 1994.Google Scholar - [32]S. Shelah,
*More on countably compact, locally countable spaces*, Israel Journal of Mathematics**62**(1988), 302–310.MathSciNetCrossRefMATHGoogle Scholar - [33]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.CrossRefGoogle Scholar - [34]S. Shelah,
*Non-reflection of the bad set for*\({\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}\to {I} _\theta }[\lambda ]\)*and pcf*, Acta Mathematica Hungarica, in press. Preprint sh:1008 in Shelah’s archive.Google Scholar - [35]L. Soukup,
*A note on Noetherian type of spaces*, arXiv:1003.3189.Google Scholar - [36]R. Telgárksy,
*C-scattered and paracompact spaces*, Fundamenta Mathematicae**73**(1971/1972), 59–74.MathSciNetGoogle Scholar - [37]S. Todorcevic,
*Directed sets and cofinal types*, Transactions of the American Mathematical Society**290**(1985), 711–723.MathSciNetCrossRefMATHGoogle Scholar