A Hausdorff compact space is metrizable if and only if it is a continuous open image of the Sorgenfrey line

TL;DR

The paper proves that a Hausdorff compact space is metrizable if and only if it is a continuous open image of the Sorgenfrey line, using properties of regular continuous open images and their subspaces.

Contribution

It establishes a new characterization of metrizable compact spaces via continuous open images of the Sorgenfrey line.

Findings

A regular continuous open image of the Sorgenfrey line with uncountable weight contains a subspace homeomorphic to the Sorgenfrey line.

This leads to the characterization of metrizable compact spaces as such images.

The proof involves analyzing subspaces and their homeomorphisms to the Sorgenfrey line.

Abstract

In this note we prove that a regular continuous open image of the Sorgenfrey line with an uncountable weight has a closed subspace that is homeomorphic to the Sorgenfrey line. As a corollary we deduce the theorem in the title.