A quasi-metrization theorem for hybrid topologies on the real line

TL;DR

This paper introduces a unifying generalization of various hybrid topologies on the real line and proves a quasi-metrization theorem for them within ZF set theory, highlighting limitations of Kofner's theorem.

Contribution

It presents a new generalization of hybrid topologies on the real line and establishes a quasi-metrization theorem without relying on the axiom of choice.

Findings

A common generalization of multiple hybrid topologies is described.

A quasi-metrization theorem for these hybrid spaces is proved in ZF.

Kofner's quasi-metrization theorem is shown to be false in certain models of ZF.

Abstract

Hybrid topologies on the real line have been studied by various authors. Among the hybrid spaces, there are also Hattori spaces. However, some of the hybrid spaces are not homeomorphic to Hattori spaces. In this article, a common generalization of at least four kinds of the hybrid topologies on the real line is described. In the absence of the axiom of choice, a quasi-metrization theorem for such hybrid spaces is proved. It is shown that Kofner's quasi-metrization theorem for generalized ordered spaces is false in every model of $\mathbf{ZF}$ in which there exists an infinite Dedekind-finite subset of the real line.