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 $([0,1]^{<\omega_1}_{\searrow 0},<_{altlex})$, 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.