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

TL;DR

This paper explores the order types of linearly ordered subsets of Baire class 1 functions, demonstrating embeddability of certain special trees and orders under various set-theoretic assumptions, and introducing the concept of compact-special trees.

Contribution

It establishes new embeddings of special Aronszajn lines and characterizes embeddability conditions for orders of size less than continuum using compact-special trees.

Findings

Special Aronszajn lines embed into Baire class 1 functions.

Under Martin's Axiom, orders without  or * embed into Baire class 1 functions.

A ZFC example shows the characterization fails for continuum-sized orders.

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 Komj\'ath 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 $\omega_1^*$. 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.