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

Elekes, Márton and Vidnyánszky, Zoltán (2017) Characterization of order types of pointwise linearly ordered families of Baire class 1 functions. ADVANCES IN MATHEMATICS, 307. pp. 559-597. ISSN 0001-8708

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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. © 2016 Elsevier Inc.

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