Continuous bijections of Borel subsets of the Sorgenfrey line on compact spaces

We prove that the topology of an uncountable Borel subset of the Sorgenfrey line is equal to the supremum of metrizable compact topologies. As a corollary we obtain that a Borel subset of the Sorgenfrey line has a weak Hausdorff compact topology if and only if it is either uncountable or countable and scattered.