Chains of baire class 1 functions and various notions of special trees |
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Authors: | Márton Elekes Juris Steprāns |
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Institution: | (1) Rényi Alfréd Institute, Reáltanoda u. 13-15, H-1053 Budapest, Hungary;(2) Department of Mathematics, York University, 4700 Keele Street, M3J 1P3 Toronto, Ontario, Canada |
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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.
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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Keywords: | |
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