GO-spaces and Noetherian spectra

TL;DR

This paper investigates the Noetherian types of ordered spaces, establishing relationships between their properties, products, and metrizability, and characterizing possible Noetherian types for compact linear orders.

Contribution

It provides new results on Noetherian types of ordered spaces, including characterizations for products, metrizability, and the spectrum of possible types for compact linear orders.

Findings

Density of product of compact linear orders never exceeds its Noetherian type.

Countable product of compact linear orders is omega_1^op-like iff it is metrizable.

Noetherian type of a compact LOTS is characterized by specific cardinalities.

Abstract

The Noetherian type of a space is the least k for which the space has a k^op-like base, i.e., a base in which no element has k-many supersets. We prove some results about Noetherian types of (generalized) ordered spaces and products thereof. For example: the density of a product of not-too-many compact linear orders never exceeds its Noetherian type, with equality possible only for singular Noetherian types; we prove a similar result for products of Lindelof GO-spaces. A countable product of compact linear orders has an omega_1^op-like base if and only if it is metrizable, and every metrizable space has an omega^op-like base. An infinite cardinal k is the Noetherian type of a compact LOTS if and only if k is not omega_1 and k is not weakly inaccessible. There is a Lindelof LOTS with Noetherian type omega_1 and there consistently is a Lindelof LOTS with weakly inaccessible Noetherian type.