Chains of baire class 1 functions and various notions of special trees
Tóm tắt
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.
Tài liệu tham khảo
J. Baumgartner, J. Malitz and W. Reinhardt,Embedding trees in the rationals, Proceedings of the National Academy of Sciences of the United States of America67 (1970), 1748–1753.
M. Elekes,Linearly ordered families of Baire 1 functions, Real Analysis Exchange27 (2001/02), 49–63.
M. Elekes and K. Kunen,Transfinite sequences of continuous and Baire class 1 functions, Proceedings of the American Mathematical Society131 (2003), 2453–2457.
T. Jech,Set Theory, Academic Press, New York, 1978.
A. S. Kechris,Classical Descriptive Set Theory, Graduate Texts in Mathematics No. 156, Springer-Verlag, Berlin, 1995.
P. Komjáth,Ordered families of Baire-2-functions, Real Analysis Exchange15 (1989/90), 442–444.
K. Kunen,Set Theory. An Introduction to Independence Proofs, Studies in Logic and the Foundations of Mathematics No. 102, North-Holland, Amsterdam, 1980.
K. Kuratowski,Topology, Academic Press, New York, 1966.
S. Shelah,Free limits of forcing and more on Aronszajn trees, Israel Journal of Mathematics38 (1981), 315–334.
S. Todorčević,Trees and linearly ordered sets, inHandbook of Set-Theoretic Topology (K. Kunen and J. E. Vaughan, eds.), North-Holland, Amsterdam, 1984, pp. 235–293.