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

我们今年的主要成果是我们在这个项目中开发的一般理论的应用。在CCA 2022上，我们给出了一些关于coPolish环（拓扑环为coPolish）的初步结果，主要结果是构造了一个从coPolish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）离散环环谱的标准构造。但是当环是非离散的时，其结果对于可计算性和基础应用来说更易于管理。在CTS 2022上，我们提出了经典描述集理论结果的一个有效版本，即在准波兰空间的拓扑上添加可数的闭集可以得到准波兰空间。我们的构造采用一个编码准波兰空间的c.e.传递关系和一个co-c.e.枚举。空间的闭子集，并输出一个用改进拓扑编码准波兰空间的c.e.传递关系。虽然经典的结果是众所周知的（拟）波兰空间，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.s selivanov的联合工作，详细分析了各种有效准波兰空间的可枚举性，取代了已知的枚举某些类域的结果。我们还证明了T1、T2和T3分离公理对应的有效拟波兰空间的子类是各（光面）共解析完备的。

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.