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

Ruiyuan Chen

Abstract

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.

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

Abstract

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" -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.
Paper Structure (32 sections, 78 theorems, 102 equations)

This paper contains 32 sections, 78 theorems, 102 equations.

Table of Contents

  1. Introduction
  2. Topological realization of relations and structures
  3. Quasi-Polish $G$-spaces
  4. Groupoid actions
  5. Proof strategy: point-free topology
  6. Organization of paper
  7. Topological preliminaries
  8. Topologies, Borel structures, and $\sigma$-topologies
  9. Quasi-Polish spaces
  10. Fiberwise topology
  11. Borel fiberwise topology
  12. Lower powerspaces
  13. Linear quantifiers
  14. Baire category for coarser topologies
  15. Polish group actions
  16. ...and 17 more sections

Key Result

theorem 1

Let $G$ be a Polish group, $X$ be a standard Borel $G$-space. For any Borel set $A \subseteq X$, the following are equivalent: Moreover, countably many $A$ obeying these conditions may be made open as in thm:intro-becker-kechris:open at the same time.

Theorems & Definitions (197)

  • theorem 1: Becker--Kechris
  • theorem 2: Sami, Hjorth
  • theorem 3: classical
  • theorem 4: characterization of "potentially open" relations
  • theorem 5: change of topology for relations
  • theorem 6: topological realization of Borel $G$-bundles of spaces
  • theorem 7: topological realization of Borel $G$-bundles of topological structures
  • definition 1
  • definition 2
  • definition 3
  • ...and 187 more