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.