Noetherian type in topological products

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 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.

The Noetherian type of the Cantor Cube of weight (Formula presented.) with the countable box topology, (Formula presented.), is shown in Section 3 to be closely related to the combinatorics of covering collections of countable subsets of (Formula presented.). We discuss the influence of principles like (Formula presented.) and Chang’s conjecture for (Formula presented.) 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 (Formula presented.) (Theorem 3.20). From this follows an absolute upper bound of ℵ₄ on the Noetherian type of (Formula presented.). The proof uses a method that was introduced by Shelah in 1993 [33].