Abstract

We consider Borel sets of finite rank $A \subseteq\Lambda^\omega$ where cardinality of $\Lambda$ is less than some uncountable regular cardinal $\mathcal{K}$. We obtain a "normal form" of A, by finding a Borel set $\Omega$, such that A and $\Omega$ continuously reduce to each other. In more technical terms: we define simple Borel operations which are homomorphic to ordinal sum, to multiplication by a countable ordinal, and to ordinal exponentiation of base $\mathcal{K}$, under the map which sends every Borel set A of finite rank to its Wadge degree.