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Chains of baire class 1 functions and various notions of special trees

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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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Authors and Affiliations

  1. Rényi Alfréd Institute, Reáltanoda u. 13-15, H-1053, Budapest, Hungary

    Márton Elekes

  2. Department of Mathematics, York University, 4700 Keele Street, M3J 1P3, Toronto, Ontario, Canada

    Juris Steprāns

Authors
  1. Márton Elekes
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  2. Juris Steprāns
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Additional information

Partially supported by Hungarian Scientific Foundation grant no. 37758, 49786 and F 43620.

The second author's research for this paper was partially supported by NSERC of Canada.

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Elekes, M., Steprāns, J. Chains of baire class 1 functions and various notions of special trees. Isr. J. Math. 151, 179–187 (2006). https://doi.org/10.1007/BF02777361

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  • DOI: https://doi.org/10.1007/BF02777361

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