论文标题
Borel group Action的结构，无点，非Hausdorff拓扑实现
Structural, point-free, non-Hausdorff topological realization of Borel groupoid actions
论文作者
论文摘要
我们扩展了贝克里斯 - 凯奇里斯拓扑实现和复杂定理，以在多个方向上进行波兰群体行动。对于波兰团体行动，我们证明了一个单一的结果，暗示了原始的贝克 - 凯奇里斯定理，以及萨米和赫乔斯的锐化，适用于Borel层次结构；通过同态性能自动连续性作用；以及“潜在开放”与“轨道方向开放”的borel bore套件的等效性。我们还表征了“潜在的” $ n $ ary关系，从而为不变的borel一阶结构提供了拓扑实现定理。然后，我们概括了群体素的作用，并证明了卢比尼的贝克尔 - 凯克里斯型定理的开放式波兰类固醇的结果，新近适应了Borel层次结构，以及对第一阶结构的光纤拓扑捆绑和捆绑的拓扑实现。 即使在波兰群体的经典案例中，我们的证明方法也是新的，并且完全基于类别量词的正式代数属性。特别是，我们不使用Metrionability或强大的Choquet游戏。因此，我们的证明在非Hausdorff上下文，开放的准派族类固醇以及更普遍的无点上下文中，对于开放的局部类固醇，我们的证据同样效果很好。
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.
