Normality in the square of the Sorgenfrey Line

TL;DR

This paper investigates the normality of the square of the Sorgenfrey line for various sets of reals, exploring conditions under which it is normal or pseudo-normal, with results depending on set-theoretic assumptions.

Contribution

It provides new examples and conditions for when the square of the Sorgenfrey line is normal or pseudo-normal, extending previous results.

Findings

If X is a Q-set, then (X[≤])^2 is normal.

For a λ-set X, (X[≤])^2 is pseudo-normal.

Assuming CH, there exists a set X where (X[≤])^2 is normal, but X is not a λ-set.

Abstract

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