A note on Noetherian type of spaces

TL;DR

This paper investigates the Noetherian type of a specific topological space derived from 2^{aleph_omega}, showing that under certain set-theoretic assumptions, its Noetherian type exceeds omega_1, thus connecting topology with advanced set theory.

Contribution

It demonstrates that under a strong form of Chang Conjecture, the Noetherian type of the space exceeds omega_1, providing a new set-theoretic insight into topological properties.

Findings

Under certain assumptions, Nt(X) > omega_1.

The space's Noetherian type depends on set-theoretic principles.

Connections between set theory and topology are established.

Abstract

The Noetherian type of a space X, Nt(X), is the least cardinal kappa such that X has a base B such that every element of the base is contained in less than kappa many elements of the base. Denote X the space obtained from 2^{aleph_omega} by declaring the G_delta sets to be open. Milovich proved that if Square_{aleph_omega} holds and (aleph_omega)^omega=aleph_{omega+1} then Nt(X)=omega_1. Answering a question of Spadaro, we show that if (aleph_omega)^omega=aleph_{omega+1} and a strong form of Chang Conjecture holds for aleph_\omega then Nt(X)>omega_1.