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Register a new userOn ordinal ranks of Baire class functions
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The theory of ordinal ranks on Baire class 1 functions developed by Kechris and Loveau was recently extended by Elekes, Kiss and Vidny{a}nszky to Baire class $xi$ functions for any countable ordinal $xigeq1$. In this paper, we answer two of the questions raised by them in their paper (Ranks on the Baire class $xi$ functions, Trans. Amer. Math. Soc. 368(2016), 8111-8143). Specifically, we show that for any countable ordinal $xigeq1,$ the ranks $beta_{xi}^{ast}$ and $gamma_{xi}^{ast}$ are essentially equivalent, and that neither of them is essentially multiplicative. Since the rank $beta$ is not essentially multiplicative, we investigate further the behavior of this rank with respect to products. We characterize the functions $f$ so that $beta(fg)leq omega^{xi}$ whenever $beta(g)leqomega^{xi}$ for any countable ordinal $xi.$
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Let $K$ be a compact metric space. A real-valued function on $K$ is said to be of Baire class one (Baire-1) if it is the pointwise limit of a sequence of continuous functions. In this paper, we study two well known ordinal indices of Baire-1 functions, the oscillation index $beta$ and the convergence index $gamma$. It is shown that these two indices are fully compatible in the following sense : a Baire-1 function $f$ satisfies $beta(f) leq omega^{xi_1} cdot omega^{xi_2}$ for some countable ordinals $xi_1$ and $xi_2$ if and only if there exists a sequence of Baire-1 functions $(f_n)$ converging to $f$ pointwise such that $sup_nbeta(f_n) leq omega^{xi_1}$ and $gamma((f_n)) leq omega^{xi_2}$. We also obtain an extension result for Baire-1 functions analogous to the Tietze Extension Theorem. Finally, it is shown that if $beta(f) leq omega^{xi_1}$ and $beta(g) leq omega^{xi_2},$ then $beta(fg) leq omega^{xi},$ where $xi=max{xi_1+xi_2, xi_2+xi_1}}.$ These results do not assume the boundedness of the functions involved.
We construct a family $(mathcal{X}_al)_{alle omega_1}$ of reflexive Banach spaces with long transfinite bases but with no unconditional basic sequences. In our spaces $mathcal{X}_al$ every bounded operator $T$ is split into its diagonal part $D_T$ and its strictly singular part $S_T$.
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W. Hurewicz proved that analytic Menger sets of reals are $sigma$-compact and that co-analytic completely Baire sets of reals are completely metrizable. It is natural to try to generalize these theorems to projective sets. This has previously been accomplished by $V = L$ for projective counterexamples, and the Axiom of Projective Determinacy for positive results. For the first problem, the first author, S. Todorcevic, and S. Tokgoz have produced a finer analysis with much weaker axioms. We produce a similar analysis for the second problem, showing the two problems are essentially equivalent. We also construct in ZFC a separable metrizable space with $omega$-th power completely Baire, yet lacking a dense completely metrizable subspace. This answers a question of Eagle and Tall in Abstract Model Theory.
We study Ramseys theorem for pairs and two colours in the context of the theory of $alpha$-large sets introduced by Ketonen and Solovay. We prove that any $2$-colouring of pairs from an $omega^{300n}$-large set admits an $omega^n$-large homogeneous set. We explain how a formalized version of this bound gives a more direct proof, and a strengthening, of the recent result of Patey and Yokoyama [Adv. Math. 330 (2018), 1034--1070] stating that Ramseys theorem for pairs and two colours is $forallSigma^0_2$-conservative over the axiomatic theory $mathsf{RCA}_0$ (recursive comprehension).
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