Chains of Baire class 1 functions and various notions of special trees

Abstract

Following Laczkovich we consider the partially ordered set $\iB_1(\RR)$ of Baire class 1 functions endowed with the pointwise order, and investigate the order types of the linearly ordered subsets. Answering a question of Komjáth and Kunen we show (in $ZFC$) that special Aronszajn lines are embeddable into $\iB_1(\RR)$. We also show that under Martin's Axiom a linearly ordered set $\mathbb{L}$ with $|\mathbb{L}|<2^{\omega }$ is embeddable into $\iB_1(\RR)$ iff $\mathbb{L}$ does not contain a copy of ${\omega }_1$ or ${\omega }_1^{\ast }$. We present a $ZFC$-example of a linear order of size $2^{\omega }$ showing that this characterisation is not valid for orders of size continuum. These results are obtained using the notion of a compact-special tree; that is, a tree that is embeddable into the class of compact subsets of the reals partially ordered under reverse inclusion. We investigate how this notion is related to the well-known notion of an $\RR$-special tree and also to some other notions of specialness.