On Properties of Nests: Some Answers and Questions

TL;DR

This paper investigates the properties of nests in topological spaces, linking order-theoretical and topological features, and explores their implications for orderability and topological groups.

Contribution

It provides new characterizations of interlocking nests, relates nests to topological properties, and proposes open questions for future research on orderability.

Findings

Characterization of interlocking nests via closed sets and lower sets

Link between nests and the structure of topological groups

Open questions on the orderability problem

Abstract

By considering nests on a given space, we explore order-theoretical and topological properties that are closely related to the structure of a nest. In particular, we see how subbases given by two dual nests can be an indicator of how close or far are the properties of the space from the structure of a linearly ordered space. Having in mind that the term interlocking nest is a key tool to a general solution of the orderability problem, we give a characterization of interlocking nest via closed sets in the Alexandroff topology and via lower sets, respectively. We also characterize bounded subsets of a given set in terms of nests and, finally, we explore the possibility of characterizing topological groups via properties of nests. All sections are followed by a number of open questions, which may give new directions to the orderability problem.