The Steel hierarchy of ordinal valued Borel mappings

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.