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

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 $([0,1{]}_{\searrow 0}^{<{\omega }_1},{<}_{altlex})$, where $[0,1{]}_{\searrow 0}^{<{\omega }_1}$ 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 $(x_{\alpha }{)}_{\alpha \le \xi },(x_{\alpha }^{\prime }{)}_{\alpha \le {\xi }^{\prime }}\in [0,1{]}_{\searrow 0}^{<{\omega }_1}$, and $\delta$ is the minimal ordinal where the two sequences differ then we say that \[ (x_α)_{α\leq ξ} <_{altlex} (x'_α)_{α\leq ξ'} \iff (δ\text{ is even and } x_δ<x'_δ) \text{ or } (δ\text{ is odd and } x_δ>x'_δ). \] Using this characterization we easily reprove all the known results and answer all the known open questions of the topic.