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. 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 ℵ_ω with the countable box topology, (2^{ℵ_ω} )δ, is shown in Section 3 to be closely related to the combinatorics of covering collections of countable subsets of ℵω. We discuss the influence of principles like ℵω and Chang’s conjecture for ℵω 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 ℵω (Theorem 3.20). From this follows an absolute upper bound of ℵ4 on the Noetherian type of (2^{ℵ_ω} )δ.