Characterization of order types of pointwise linearly ordered families of Baire class 1 functions

TL;DR

This paper completely characterizes the order types of linearly ordered families of Baire class 1 functions on uncountable Polish spaces, linking them to a specific transfinite sequence order with an alternating lexicographical structure.

Contribution

It provides a complete solution to Laczkovich's problem by characterizing order types via a new order structure involving transfinite sequences and alternating lexicographical order.

Findings

Characterizes order types of Baire class 1 functions as subsets of a specific transfinite order.

Introduces the alternating lexicographical order on decreasing transfinite sequences.

Reproves known results and resolves open questions in the field.

Abstract

In the 1970s M. Laczkovich posed the following problem: Let $\mathcal{B}_1(X)$ denote the set of Baire class $1$ functions defined on an uncountable Polish space $X$ equipped with the pointwise ordering. \[\text{Characterize the order types of the linearly ordered subsets of $\mathcal{B}_1(X)$.} \]The main result of the present paper is a complete solution to this problem. We prove that a linear order is isomorphic to a linearly ordered family of Baire class $1$ functions iff it is isomorphic to a subset of the following linear order that we call $([0,1]^{<\omega_1}_{\searrow 0},<_{altlex})$, where $[0,1]^{<\omega_1}_{\searrow 0}$ is the set of strictly decreasing transfinite sequences of reals in $[0, 1]$ with last element $0$, and $<_{altlex}$, the so called \emph{alternating lexicographical ordering}, is defined as follows: if $(x_\alpha)_{\alpha\leq \xi}, (x'_\alpha)_{\alpha\leq \xi'} \in [0,1]^{<\omega_1}_{\searrow 0}$, and $\delta$ is the minimal ordinal where the two sequences differ then we say that \[ (x_\alpha)_{\alpha\leq \xi} <_{altlex} (x'_\alpha)_{\alpha\leq \xi'} \iff (\delta \text{ is even and } x_{\delta}<x'_{\delta}) \text{ or } (\delta \text{ is odd and } x_{\delta}>x'_{\delta}). \] Using this characterization we easily reprove all the known results and answer all the known open questions of the topic.