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发展的一般理论的应用。在CCA 2022上，我们提出了一些关于coPolish环的初步结果。（拓扑为coPolish的拓扑环），主要结果是从coPolish环和连续环同态范畴到拟Polish环范畴的一个逆变函子的构造。波兰空间，符合环的谱的标准构造，（可数）离散环，但当环是非离散的时，会产生对于可计算性和基础应用更易于管理的空间。在CTS 2022上，我们给出了经典描述集合论结果的一个有效版本：将可数个闭集加到拟波兰空间的拓扑上，得到拟波兰空间。我们的建筑需要一个c.e.传递关系编码的准波兰空间和计数的co-c. e。闭子集的空间，并输出一个c.e.传递关系编码的准波兰空间与细化拓扑。虽然经典的结果是众所周知的（准）波兰空间，据我们所知，这是第一个有效的证明。在CiE 2022，我们提出了与T.Kihara和V.Selivanov的联合工作，给出了详细的分析各类有效的准波兰空间的可枚举性，这取代了已知的结果枚举某些类别的域。我们还证明了对应于T1、T2和T3分离公理的有效拟波兰空间的子类都是（光面）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.