Theory and applications of Stone-duality for quasi-Polish spaces

我们今年的主要结果是我们在这个项目中发展的一般理论的应用。在CCA 2022上，我们给出了关于余波兰环(拓扑环的拓扑是余波兰环)的一些初步结果，主要结果是构造了一个从余波兰环和连续环同态范畴到准波兰空间范畴的逆变函子，它符合(可数)离散环的环谱的标准结构，但当环是非离散的时，得到了更易于计算和基础应用的空间。我们给出了经典描述集合论结果的一个有效版本，即在拟Polish空间的拓扑上加上可数个闭集就得到了拟Polish空间。我们的建筑需要C.E。编码拟Polish空间的传递关系与co-C.E.空间的闭子集，并输出C.E.用精化拓扑编码拟波兰空间的传递关系。虽然经典的结果对于(准)波兰空间是众所周知的，但据我们所知，这是第一个有效的证明。在2022年，我们与T.Kihara和V.Selivanov合作，给出了各种有效的拟波兰空间的可数性的详细分析，它取代了已知的关于计数某些类区域的结果。我们还证明了对应于T1、T2和T3分离公理的有效拟Polish空间的子类都是(LightFace)余分析完备的。

Our main results this year were applications of the general theory that we developed during this project.At CCA 2022, we presented some preliminary results about coPolish rings (topological rings whose topology is coPolish), the main result being the construction of a contravariant functor from the category of coPolish rings and continuous ring homomorphisms to the category of quasi-Polish spaces that agrees with the standard construction of the spectrum of a ring for (countable) discrete rings, but results in spaces that are more manageable for computability and foundational applications when the ring is non-discrete.At CTS 2022, we presented an effective version of the classical descriptive set theory result that adding countably many closed sets to the topology of a quasi-Polish space results in a quasi-Polish space. Our construction takes a c.e. transitive relation encoding a quasi-Polish space and an enumeration of co-c.e. closed subsets of the space and outputs a c.e. transitive relation encoding the quasi-Polish space with the refined topology. Although the classical result is well-known for (quasi-) Polish spaces, to our knowledge this is the first effective proof.At CiE 2022, we presented joint work with T.Kihara and V.Selivanov that gives a detailed analysis of the enumerability of various classes of effective quasi-Polish spaces, which supersedes known results on enumerating certain classes of domains. We also showed that the subclasses of effective quasi-Polish spaces corresponding to the T1,T2, and T3 separation axioms are each (lightface) coanalytic complete.