K-Triviality of Closed Sets and Continuous Functions

Abstract

We investigate the notion of K-triviality for closed sets and continuous functions in 2ℕ. For every K-trivial degree d, there exists a closed set of degree d and a continuous function of degree d. Every K-trivial closed set contains a K-trivial real. There exists a K-trivial Π10 class with no computable elements. A closed set is K-trivial if and only if it is the set of zeroes of some K-trivial continuous function. We give a density result for the Medvedev degrees of K-trivial Π10 sets. If W ≤TA′, then W can compute a path through every A′-decidable random closed set if and only if W ≡TA′.