Noetherian types of homogeneous compacta and dyadic compacta

TL;DR

This paper investigates the Noetherian type of homogeneous compact spaces, establishing bounds and properties, including that homogeneous dyadic compacta have Noetherian type ω and exploring implications under GCH.

Contribution

It provides new bounds on Noetherian types of homogeneous compacta and characterizes their local base structures, extending understanding of their topological properties.

Findings

Homogeneous dyadic compacta have Noetherian type ω.

Under GCH, points in homogeneous compacta have c(X)-like local bases.

If all points have well-quasiordered local bases, some point has a countable local π-base.

Abstract

The Noetherian type of a space is the least $\kappa$ such that it has a base that is $\kappa$-like with respect to containment. Just as all known homogeneous compacta have cellularity at most $2^\omega$, they satisfy similar upper bounds in terms of Noetherian type and related cardinal functions. We prove these and many other results about these cardinal functions. For example, every homogeneous dyadic compactum has Noetherian type $\omega$. Assuming GCH, every point in a homogeneous compactum $X$ has a local base that is $c(X)$-like with respect to containment. If every point in a compactum has a well-quasiordered local base, then some point has a countable local $\pi$-base.