Normality in the square of the Sorgenfrey Line
Paul Szeptycki, Hongwei Wen
TL;DR
The paper investigates when the Sorgenfrey square $(X[\leq])^2$ is normal for subsets $X\subseteq\mathbb{R}$ and how this relates to $Q$-sets and $\lambda$-sets. It establishes a partial converse: if $(X[\leq])^2$ is normal and a strictly decreasing map $f:X\to X$ exists, then $X$ must be a $Q$-set, and a weaker version yields $X\setminus C$ is $Q$-set for some countable $C$; under CH these implications fail in general, with uncountable $X$ that are not $Q$-sets yet have normal $(X[\leq])^2$ and even constructions of entangled/2-entangled-like behavior. The paper also proves that $\lambda$-sets force $(X[\leq])^2$ to be pseudo-normal, and provides a partial converse stating that pseudo-normality together with a strictly decreasing Euclidean homeomorphism implies $X$ is a $\lambda$-set. These results illuminate the delicate interaction between order-theoretic, topological, and set-theoretic assumptions in Sorgenfrey-type squares and raise questions about the consistency of various converses.
Abstract
We consider sets of reals $X$ endowed with the Sorgenfrey lower limit topology denoted $X[\leq]$. Przymusiński 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 $λ$ set, then $(X[\leq])^2$ is pseudo-normal but assuming CH there is an $X$ concentrated on a countable dense subset (so not a $λ$-set) but still $(X[\leq])^2$ is normal.