J. Lutz and N. Lutz (2017) have recently proven a point-to-set principle for Euclidean and Cantor spaces. This result is a characterization of classical Hausdorff dimension in terms of relativized effective dimension. This implies that geometric measure results regarding Hausdorff dimension can be shown using only effective methods. Several interesting classical results have already been proven using this principle (N. Lutz and Stull 2017, N. Lutz 2017).

We present here a point-to-set principle for any separable metric space. We will first introduce an effectivization of dimension in terms of Kolmogorov complexity that is valid for any separable space, and then show that the classical Hausdorff dimension of any set is the minimum on all oracles of the relativized effective dimension. We expect that better Hausdorff dimension bounds will be proven as consequences of this theorem.