Noetherian type in topological products

TL;DR

This paper investigates the behavior of the Noetherian type in topological products, revealing its non-monotonicity, preservation in certain classes, and connections to set-theoretic principles and PCF theory.

Contribution

It demonstrates that Noetherian type can decrease under products, explores its preservation in compact spaces, and links its value to advanced set-theoretic concepts like PCF theory and large cardinals.

Findings

Existence of spaces where Nt(X×Y) < min{Nt(X), Nt(Y)}

Preservation of Nt in certain compact spaces under specific operations

Bound of  on Noetherian type of ^{\u0015}_ ext{omega}

Abstract

The cardinal invariant "Noetherian type" of a topological space $X$ (Nt(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 $Nt(X \times Y) < \min\{Nt(X), Nt(Y)\}$. 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 $(\aleph_4,\aleph_1)$-sparse covering family of countable subsets of $\aleph_\omega$. From this follows an absolute upper bound of $\aleph_4$ on the Noetherian type of $(2^{\aleph_\omega})_\delta$. The proof uses ideas from Shelah's proof that if $\kappa^+ <\lambda$ then his ideal $I[\lambda]$ contains a stationary set consisting of points of cofinality $\kappa$.