A Hausdorff compact space is metrizable if and only if it is a continuous open image of the Sorgenfrey line
arXiv:2110.12808
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.