  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 $\mathbb{R}^n$. 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
fractal dimensions--computability-theoretic versions of classical Hausdorff and
packing dimensions that assign dimensions $\dim(x)$ and $\textrm{Dim}(x)$ to
individual points $x\in X$--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. (For a
concrete computational example, the stages $E_0, E_1, E_2, \ldots$ used to
construct a self-similar fractal $E$ in the plane are elements of the
hyperspace of the plane, and they converge to $E$ in the hyperspace.) Our third
main result, proven via our extended point-to-set principle, states that, under
a wide variety of gauge families, the classical packing dimension agrees with
the classical upper Minkowski dimension on all hyperspaces of compact sets. We
use this theorem to give, for all sets $E$ that are analytic, i.e.,
$\mathbf{\Sigma}^1_1$, a tight bound on the packing dimension of the hyperspace
of $E$ in terms of the packing dimension of $E$ itself.
