, Volume 202, Issue 1, pp 195-225
Date: 23 Jul 2014

Noetherian type in topological products

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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.

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 (ℵ4, ℵ1)-sparse covering family of countable subsets of \({\aleph _\omega }\) (Theorem 3.20). From this follows an absolute upper bound of ℵ4 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 INdAM-Cofund 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.