The Steel hierarchy of ordinal valued Borel mappings

Abstract

AbstractGiven well ordered countable sets of the form Λϕ, we consider Borel mappings from Λϕω with countable image inside the ordinals. The ordinals and Λϕω are respectively equipped with the discrete topology and the product of the discrete topology on Λϕ. The Steel well-ordering on such mappings is denned by ϕ ≤ ψ iff there exists a continuous function f such that ϕ(x) ≤ ψof(x) holds for any x ϵ Λϕω. It induces a hierarchy of mappings which we give a complete description of. We provide, for each ordinal α, a mapping 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.