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

我们今年的主要成果是我们在该项目中开发的一般理论的应用。在 CCA 2022 上，我们提出了关于 coPolish 环（拓扑为 coPolish 的拓扑环）的一些初步结果，主要结果是从 coPolish 环和连续环同态类别到准波兰空间类别的逆变函子的构造，该构造与（可数）离散环的环谱的标准构造一致，但产生的空间更多当环是非离散时，可计算性和基础应用是可管理的。在 CTS 2022 上，我们提出了经典描述集合论结果的有效版本，即向准波兰空间的拓扑中添加可数个闭集会产生准波兰空间。我们的建设需要一个 c.e.传递关系编码准波兰空间和 co-c.e 的枚举。空间的闭合子集并输出 c.e.传递关系用精致的拓扑编码准波兰空间。尽管（准）波兰空间的经典结果众所周知，但据我们所知，这是第一个有效的证明。在 CiE 2022 上，我们展示了与 T.Kihara 和 V.Selivanov 的联合工作，详细分析了各种有效准波兰空间类别的可枚举性，这取代了枚举某些类别域的已知结果。我们还表明，与 T1、T2 和 T3 分离公理相对应的有效准波兰空间的子类都是（lightface）协同解析完备的。

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.