K-triviality in computable metric spaces

摘要

A point x in a computable metric space is called K-trivial if for each positive rational δ there is an approximation p at distance at most δ from x such that the pair p, δ is highly compressible in the sense that K(p, δ) ≤ K(δ) + O(1). We show that this local definition is equivalent to the point having a Cauchy name that is K-trivial when viewed as a function from ? to ?. We use this to transfer known results on K-triviality for functions to the more general setting of metric spaces. For instance, we show that each computable Polish space without isolated points contains an incomputable K-trivial point.