Infinitary Noetherian Constructions II. Transfinite Words and the Regular Subword Topology

TL;DR

This paper investigates the properties of transfinite words over Noetherian spaces, demonstrating that these spaces are also Noetherian under a specific topology and analyzing their structural characteristics.

Contribution

It introduces the regular subword topology for transfinite words and characterizes the sobrification, specialization order, and bounds on dimension and stature.

Findings

Spaces of transfinite words over Noetherian spaces are Noetherian under the regular subword topology.

The sobrification and specialization order of these spaces are characterized.

Upper bounds on the dimension and stature of the space are provided.

Abstract

We show that the spaces of transfinite words, namely ordinal-indexed words, over a Noetherian space, is also Noetherian, under a natural topology which we call the regular subword topology. We characterize its sobrification and its specialization ordering, and we give an upper bound on its dimension and on its stature.