$K$-triviality in computable metric spaces

Abstract:

A point $x$ in a computable metric space is called $K$-trivial if for each positive rational $\delta$ there is an approximation $p$ at distance at most $\delta$ from $x$ such that the pair $p, \delta$ is highly compressible in the sense that $K(p, \delta ) \le K(\delta ) + O(1)$. 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 $\mathbb {N}$ to $\mathbb {N}$. 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.