Splitting families and the Noetherian type of $\beta\omega-\omega$

TL;DR

This paper investigates the Noetherian type of the space eta extomega-, revealing its independence from certain set-theoretic assumptions and its relation to splitting families and other cardinal invariants.

Contribution

It extends previous results by proving independence results about the Noetherian type of eta extomega- and its powers, linking it to splitting families and set-theoretic invariants.

Findings

The Noetherian type is never less than the splitting number.

It can consistently equal  extomega_1, 2^ extomega, or values between them.

The Noetherian type can be less than the additivity of the meager ideal.

Abstract

Extending some results of Malykhin, we prove several independence results about base properties of $\beta\omega-\omega$ and its powers, especially the Noetherian type $Nt(\beta\omega-\omega)$, the least $\kappa$ for which $\beta\omega-\omega$ has a base that is $\kappa$-like with respect to containment. For example, $Nt(\beta\omega-\omega)$ is never less than the splitting number, but can consistently be that $\omega_1$, $2^\omega$, $(2^\omega)^+$, or strictly between $\omega_1$ and $2^\omega$. $Nt(\beta\omega-\omega)$ is also consistently less than the additivity of the meager ideal. $Nt(\beta\omega-\omega)$ is closely related to the existence of special kinds of splitting families.