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March 2003 The Steel hierarchy of ordinal valued Borel mappings
J. Duparc
J. Symbolic Logic 68(1): 187-234 (March 2003). DOI: 10.2178/jsl/1045861511

Abstract

Given well ordered countable sets of the form $\lamphi$, we consider Borel mappings from $\lamphiom$ with countable image inside the ordinals. The ordinals and $\lamphiom$ are respectively equipped with the discrete topology and the product of the discrete topology on $\lamphi$. The Steel well-ordering on such mappings is defined by $\phi\minf\psi$ iff there exists a continuous function $f$ such that $\phi(x)\leq\psi\circ f(x)$ holds for any $x\in\lamphiom$. It induces a hierarchy of mappings which we give a complete description of. We provide, for each ordinal $\alpha$, a mapping $\T{\alpha}$ whose rank is precisely $\alpha$ in this hierarchy and we also compute the height of the hierarchy restricted to mappings with image bounded by $\alpha$. These mappings being viewed as partitions of the reals, there is, in most cases, a unique distinguished element of the partition. We analyze the relation between its topological complexity and the rank of the mapping in this hierarchy.

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J. Duparc. "The Steel hierarchy of ordinal valued Borel mappings." J. Symbolic Logic 68 (1) 187 - 234, March 2003. https://doi.org/10.2178/jsl/1045861511

Information

Published: March 2003
First available in Project Euclid: 21 February 2003

zbMATH: 1050.03032
MathSciNet: MR1959317
Digital Object Identifier: 10.2178/jsl/1045861511

Rights: Copyright © 2003 Association for Symbolic Logic

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Vol.68 • No. 1 • March 2003
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