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 B1(X) denote the set of Baire class 1 functions defined on an uncountable Polish space X equipped with the pointwise ordering.Characterize the order types of the linearly orderedsubsets of B1(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]↘0<ω1,<altlex), where [0,1]↘0<ω1 is the set of strictly decreasing transfinite sequences of reals in [0,1] with last element 0, and <altlex, the so called alternating lexicographical ordering, is defined as follows: if (xα)α≤ξ,(xα′)α≤ξ′∈[0,1]↘0<ω1 are distinct, and δ is the minimal ordinal where the two sequences differ then we say that(xα)α≤ξ<altlex(xα′)α≤ξ′⇔(δ is even and xδ<xδ′) or (δ is odd and xδ>xδ′).Using this characterization we easily reprove all the known results and answer all the known open questions of the topic.