Countable and finitary reductions on equivalence relations

Inspired by the very successful study of Borel equivalence relations under Borel reducibility in descriptive set theory and equivalence relations on $\omega $ under computable reducibility in computability theory, R. Miller defined a family of reducibility notions. Defined on equivalence relations on Baire space or Cantor space, these reducibilities are required to succeed (uniformly) on all finite or countable subsets of the whole space. In this paper, we combine methods from computability theory and descriptive set theory to study equivalence relations under these reductions. In particular, we show existence and non-existence of complete equivalence relations in various settings.