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The Steel Hierarchy of Ordinal Valued Borel Mappings

J. Duparc
The Journal of Symbolic Logic
Vol. 68, No. 1 (Mar., 2003), pp. 187-234
Stable URL: http://www.jstor.org/stable/4147739
Page Count: 48
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Since scans are not currently available to screen readers, please contact JSTOR User Support for access. We'll provide a PDF copy for your screen reader.
The Steel Hierarchy of Ordinal Valued Borel Mappings
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Abstract

Given well ordered countable sets of the form $\Lambda_\phi$, we consider Borel mappings from $\Lambda_\phi^\omega$ with countable image inside the ordinals. The ordinals and $\Lambda_\phi^\omega$ are respectively equipped with the discrete topology and the product of the discrete topology on $\Lambda_\phi$. The Steel well-ordering on such mappings is defined by $\phi \leq \psi$ iff there exists a continuous function f such that $\phi(x) \leq \psi \circ f(x)$ holds for any $x \in \Lambda_\phi^\omega$. It induces a hierarchy of mappings which we give a complete description of. We provide, for each ordinal α, a mapping T(α) whose rank is precisely α in this hierarchy and we also compute the height of the hierarchy restricted to mappings with image bounded by α. 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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