Extending the reach of the point-to-set principle

Abstract

The point-to-set principle of J. Lutz and N. Lutz (2018) has recently enabled the theory of computing to be used to answer open questions about fractal geometry in Euclidean spaces Rn. These are classical questions, meaning that their statements do not involve computation or related aspects of logic.In this paper we extend the reach of the point-to-set principle from Euclidean spaces to arbitrary separable metric spaces X. We first extend two algorithmic dimensions—computability-theoretic versions of classical Hausdorff and packing dimensions that assign dimensions dim⁡(x) and Dim(x) to individual pointsx∈X—to arbitrary separable metric spaces and to arbitrary gauge families. Our first two main results then extend the point-to-set principle to arbitrary separable metric spaces and to a large class of gauge families.We demonstrate the power of our extended point-to-set principle by using it to prove new theorems about classical fractal dimensions in hyperspaces.