Open images of the Sorgenfrey line

TL;DR

This paper characterizes the Hausdorff continuous open images of the Sorgenfrey line as spaces with a Sorgenfrey base and shows certain compact spaces cannot be such images, highlighting limitations in their topological relationships.

Contribution

It provides a complete description of Hausdorff continuous open images of the Sorgenfrey line and establishes non-existence results for certain compact spaces as such images.

Findings

Hausdorff continuous open images have Sorgenfrey bases

Double-arrow space is not a continuous open image of the Sorgenfrey line

No compact Hausdorff space containing a Sorgenfrey line is its open image

Abstract

We give a description of Hausdorff continuous open images of the Sorgenfrey line: these are precisely those spaces that have a Sorgenfrey base. Using this description we prove that no Hausdorff compact space that contains a copy of the Sorgenfrey line is a continuous open image of it; in particular the double-arrow space is not a continuous open image of the Sorgenfrey line.