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

Details Chains of Baire class 1 functions and various notions of special trees Chains of Baire class 1 functions and various notions of special trees

2011-09-24

arxiv.org/abs/1109.5283v2

Israel J. Math. 151 (2006), 179-187

Mathematics

Logic

Scientific paper

Following Laczkovich we consider the partially ordered set $\iB_1(\RR)$ of Baire class 1 functions endowed with the pointwise order, and investigate the order types of the linearly ordered subsets. Answering a question of Komj\'ath and Kunen we show (in $ZFC$) that special Aronszajn lines are embeddable into $\iB_1(\RR)$. We also show that under Martin's Axiom a linearly ordered set $\mathbb{L}$ with $|\mathbb{L}|<2^\omega$ is embeddable into $\iB_1(\RR)$ iff $\mathbb{L}$ does not contain a copy of $\omega_1$ or $\omega_1^*$. We present a $ZFC$-example of a linear order of size $2^\omega$ 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 $\RR$-special tree and also to some other notions of specialness.

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