On two topological cardinal invariants of an order-theoretic flavour

TL;DR

This paper investigates two topological cardinal invariants, the Noetherian type and Noetherian π-type, analyzing their behavior in specific spaces like κ-Suslin lines and box products, and connecting them to PCF theory.

Contribution

It determines the Noetherian π-type for κ-Suslin lines up to the first singular cardinal and links these invariants to PCF theory and hypotheses like Chang's Conjecture.

Findings

Noetherian π-type of κ-Suslin lines is determined up to the first singular cardinal.

A consequence of Chang's Conjecture is shown to influence the Noetherian type of certain box products.

Connections are established between PCF theory and the Noetherian type of Pixley-Roy hyperspaces.

Abstract

Noetherian type and Noetherian $\pi$-type are two cardinal functions which were introduced by Peregudov in 1997, capturing some properties studied earlier by the Russian School. Their behavior has been shown to be akin to that of the \emph{cellularity}, that is the supremum of the sizes of pairwise disjoint non-empty open sets in a topological space. Building on that analogy, we study the Noetherian $\pi$-type of $\kappa$-Suslin Lines, and we are able to determine it for every $\kappa$ up to the first singular cardinal. We then prove a consequence of Chang's Conjecture for $\aleph_\omega$ regarding the Noetherian type of countably supported box products which generalizes a result of Lajos Soukup. We finish with a connection between PCF theory and the Noetherian type of certain Pixley-Roy hyperspaces.