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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 $xi.$
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 function
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 ac
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 s