2018 Fiscal Year Research-status Report
Theory and applications of Stone-duality for quasi-Polish spaces
| Project/Area Number |
18K11166
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| Research Institution | Kyoto University |
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Principal Investigator |
ディブレクト マシュー 京都大学, 人間・環境学研究科, 特定講師 (20623599)
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| Project Period (FY) |
2018-04-01 – 2023-03-31
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| Keywords | quasi-Polish space / Stone-duality / topology / frames / descriptive set theory / domain theory / computable analysis / geometric logic |
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| Outline of Annual Research Achievements |
One of the goals of this research is to better understand Stone-duality for frames (i.e., complete lattices satisfying an infinite distributivity law) that can be naturally equipped with a quasi-Polish topology such that the frame operations are continuous functions. This research intersects the fields of domain theory, point-free topology, and descriptive set theory, and it is important to understand how the proof techniques from these fields can be applied within our framework.
During the first year of this project, we have looked at various characterizations of quasi-Polish frames. More generally, we obtained some important technical results concerning the descriptive complexity of natural classes of topological semi-lattices and frames. For example, we showed that if a topological semi-lattice is homeomorphic to the semi-lattice of compact subsets of a countably based co-analytic sober space (ordered by reverse inclusion and given the Scott-topology), then the semi-lattice is analytic (i.e., it is a continuous image of Baire space) if and only if it is quasi-Polish. A similar result applies to the complexity of the frame of open subsets (equipped with the Scott-topology) of a certain class of topological spaces. These results bring together important concepts and results from both domain theory and descriptive set theory, and are useful for characterizing frames that are quasi-Polish with respect to the Scott-topology.
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| Current Status of Research Progress |
Current Status of Research Progress
2: Research has progressed on the whole more than it was originally planned.
Reason
The characterization mentioned above generalizes to show that for a certain class of topological semilattices (specifically, countably based coanalytic sober algebras of the upper powerspace monad), being analytic is equivalent to being quasi-Polish. This result was an important step forward to better understanding quasi-Polish frames and topological semilattices in general.
In addition to making progress on characterizing quasi-Polish frames, we also made some progress on characterizing consonant sober spaces (i.e., those spaces for which the Scott-topology on its frame of opens has a basis of Scott-open filters) and found new criteria for spaces to have spatial localic products.
The results on the descriptive complexity of topological lattices mentioned above were obtained by a close analysis of a countable space called S0. This is one of four "canonical" non-quasi-Polish spaces, as shown by a generalization of Hurewicz's result showing that the space of rationals is the "canonical" non-Polish separable metrizable space. Further analysis of S0 showed that it is not consonant, and that the localic product of S0 with itself is not spatial. Similar negative results for the rationals are well-known, and played an important role in characterizing consonance and the spatiality of localic products among separable metrizable spaces. Therefore, this technical result is also significant because it suggests S0 will play a role similar to the rationals in a more general characterization of consonance and spatial localic products among countably based spaces.
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| Strategy for Future Research Activity |
Our partial results for S0 already imply that quasi-Polish spaces form the largest full subcategory C of countably based co-analytic sober spaces such that (1) C contains all finite spaces, (2) the functor from spaces to locales preserves products when restricted to C, and (3) C is closed under countable limits. We are currently investigating whether our results for S0 can be generalized to provide a more complete characterization of consonance and spatial localic products, in particular, to what extent criterion (3) can be weakened.
We have also been investigating a suitable definition of "computable" quasi-Polish space, and we are interested in potential interactions with computability theory and computable model theory. We also plan to further investigate the connections between formal topology (a constructive and predicative approach to point-free topology) and the concepts arising when effectivizing the theory of quasi-Polish spaces.
We also plan to continue making progress on characterizing the sober spaces whose frame of open sets is quasi-Polish. The case of regular Hausdorff sequential spaces has already been completely settled. Natural examples of these spaces exist in topological algebra (specifically within the duality theory of Banach spaces and in the theory of free topological groups). We are now looking at whether or not non-sequential regular Hausdorff spaces exist that have a quasi-Polish frame of open sets, but this appears to be a difficult problem.
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