Abstract
Following Laczkovich we consider the partially ordered setB 1(ℝ) of Baire class 1 functions endowed with the pointwise order, and investigate the order types of the linearly ordered subsets. Answering a question of Komjáth and Kunen we show (inZFC) that special Aronszajn lines are embeddable intoB 1(ℝ). We also show that under Martin's Axiom a linearly ordered set ℒ with |ℒ| < 2ω is embeddable intoB 1(ℝ) iff ℒ does not contain a copy of ω1 or ω 1* . We present aZFC example of a linear order of size 2ω showing that this characterisation is not valid for orders of size continuum.These results are obtained using the notion of a compact-special tree; that is, a tree that is embeddable into the class of compact subsets of the reals partially ordered under reverse inclusion. We investigate how this notion is related to the well-known notion of an ℝ-special tree and also to some other notions of specialness.
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