Normality in the square of the Sorgenfrey Line

Abstract

We consider sets of reals $X$ endowed with the Sorgenfrey lower limit topology denoted $X[\le ]$. Przymusiński proved that if $X$ is a $Q$-set then $(X[\le ]{)}^2$ is normal. While the converse is not in general true we consider examples of sets of the reals for which $(X[\le ]{)}^2$ is normal or just pseudo-normal. For example, if $X$ is a $\hat{I}»$ set, then $(X[\le ]{)}^2$ is pseudo-normal but assuming CH there is an $X$ concentrated on a countable dense subset (so not a $\hat{I}»$-set) but still $(X[\le ]{)}^2$ is normal.