2019 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 / topology / locale theory / domain theory |
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| Outline of Annual Research Achievements |
We made much progress on understanding a new characterization of quasi-Polish spaces as spaces of ideals of a transitive binary relation on a countable set. This characterization was published in the joint paper with A. Pauly and M. Schroder, where it was used as part of a characterization of "computable" quasi-Polish spaces. The characterization is closely related to well-known constructions of continuous domains in terms of abstract bases. Since the countable set underlying the relation represents a collection of basic open subsets, this characterization is also related to some pointfree approaches to topology, such as formal topology.
In the joint paper with J. Goubault-Larrecq, J. Xiaodong, and L. Zhenchao we investigated LCS-complete spaces, which are generalizations of quasi-Polish spaces beyond the realm of countably based spaces. They can be viewed dually as quotient frames of continuous distributive lattices, where there is a certain countability restriction on the congruence relation defining the quotient. We showed these spaces have several nice properties, such as being Wilker spaces, being Baire spaces, being consonant, and that every continuous valuation on an LCS-complete space extends to a Borel measure on the space.
We also extended some of our results on characterizing the countably based consonant spaces, and the countably based spaces which have spatial localic products.
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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 of quasi-Polish spaces in terms of spaces of ideals has been helpful in clarifying the connections between these spaces and computable topology, domain theory, and pointfree topology. The characterization itself is simple and natural, but we have also found that the basic theory (continuous functions, power spaces, etc.) can be developed smoothly in terms of spaces of ideals.
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| Strategy for Future Research Activity |
The characterization in terms of spaces of ideals gives alternative constructions of quasi-Polish powerspaces based on similar constructions from domain theory. We plan to see if these new characterizations of the powerspaces can provide any more insights into the structure of quasi-Polish algebras of the powerspace monads, and quasi-Polish frames in particular. It also provides new perspectives on investigating the coalgebras of the powerspace functors and their relationship with models of modal logic. We are also interested in further investigating the computability aspects of these constructions, and perhaps begin some reverse mathematical investigations.
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| Causes of Carryover |
Some funds remained because some workshops and other meetings were canceled near the end of the fiscal year. The funds will be used to attend the workshops when they are rescheduled.
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