Linearly Ordered Families of Baire 1 Functions

We consider the set of Baire 1 functions endowed with the pointwise partial ordering and investigate the structure of the linearly ordered subsets.

💡 Research Summary

This paper investigates the linear (total) subsets of the partially ordered set F consisting of real‑valued Baire‑1 functions on an arbitrary set X, ordered pointwise: f ≤ g iff f(x) ≤ g(x) for every x∈X. While the situation is completely understood for continuous functions (Baire‑0) – where a linear subset is representable exactly when it is order‑isomorphic to a subset of the real line – and for higher Baire classes (α > 1) – where representability is independent of ZFC – the case α = 1 remains partially open. Two classical restrictions are recalled: (i) Kuratowski’s theorem forbids an increasing (or decreasing) ω₁‑sequence of Baire‑1 functions, so ω₁ itself is not representable; (ii) Komjáth’s theorem shows that no Souslin line can be represented, and the existence of Souslin lines is independent of ZFC.

The author’s main goal is to show that, despite these limitations, a surprisingly rich family of order types can be represented by Baire‑1 functions. The strategy consists of two parts: (a) identifying operations on ordered sets that preserve representability, and (b) using these operations to build complex order types from simpler ones.

Preservation operations.

• Duplication: For an ordered set X, the lexicographic product X × {0,1} (called the duplication of X) is shown to be representable whenever X is. The proof uses a Peano curve P: