  We extend the Becker--Kechris topological realization and change-of-topology
theorems for Polish group actions in several directions. For Polish group
actions, we prove a single result that implies the original Becker--Kechris
theorems, as well as Sami's and Hjorth's sharpenings adapted levelwise to the
Borel hierarchy; automatic continuity of Borel actions via homeomorphisms; and
the equivalence of "potentially open" versus "orbitwise open" Borel sets. We
also characterize "potentially open" $n$-ary relations, thus yielding a
topological realization theorem for invariant Borel first-order structures. We
then generalize to groupoid actions, and prove a result subsuming Lupini's
Becker--Kechris-type theorems for open Polish groupoids, newly adapted to the
Borel hierarchy, as well as topological realizations of actions on fiberwise
topological bundles and bundles of first-order structures.
  Our proof method is new even in the classical case of Polish groups, and is
based entirely on formal algebraic properties of category quantifiers; in
particular, we make no use of either metrizability or the strong Choquet game.
Consequently, our proofs work equally well in the non-Hausdorff context, for
open quasi-Polish groupoids, and more generally in the point-free context, for
open localic groupoids.
