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

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. $$\text{Characterize the order types of the linearly ordered subsets of }{\mathcal{B}}_1(X)\text{.}$$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 $(x_\alpha){\alpha\leq \xi}, (x’\alpha){\alpha\leq \xi’} \in [0,1]^{<\omega_1}{\searrow 0}$,and$\delta$ is the minimal ordinal where the two sequences differ then we say that $$(x_{\alpha }{)}_{\alpha \le \xi }{<}_{altlex}(x_{\alpha }^{\mathrm{\prime }}{)}_{\alpha \le {\xi }^{\mathrm{\prime }}}⟺(\delta \text{ is even and }{x}_{\delta }<x_{\delta }^{\mathrm{\prime }})\text{ or }(\delta \text{ is odd and }{x}_{\delta }>x_{\delta }^{\mathrm{\prime }})\mathrm{.}$$ Using this characterization we easily reprove all the known results and answer all the known open questions of the topic.