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

Structural, point-free, non-Hausdorff topological realization of Borel groupoid actions