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DOI: 10.4230/LIPIcs.CSL.2017.16
URN: urn:nbn:de:0030-drops-76988
URL: https://drops.dagstuhl.de/opus/volltexte/2017/7698/
de Brecht, Matthew ; Pauly, Arno
Noetherian Quasi-Polish spaces
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Abstract
In the presence of suitable power spaces, compactness of X can be characterized as the singleton {X} being open in the space O(X) of open subsets of X. Equivalently, this means that universal quantification over a compact space preserves open predicates.Using the language of represented spaces, one can make sense of notions such as a Sigma^0_2-subset of the space of Sigma^0_2-subsets of a given space. This suggests higher-order analogues to compactness: We can, e.g., investigate the spaces X where {X} is a Delta^0_2-subset of the space of Delta^0_2-subsets of X. Call this notion nabla-compactness. As Delta^0_2 is self-dual, we find that both universal and existential quantifier over nabla-compact spaces preserve Delta^0_2 predicates.
Recall that a space is called Noetherian iff every subset is compact. Within the setting of Quasi-Polish spaces, we can fully characterize the nabla-compact spaces: A Quasi-Polish space is Noetherian iff it is nabla-compact. Note that the restriction to Quasi-Polish spaces is sufficiently general to include plenty of examples.
BibTeX - Entry
@InProceedings{debrecht_et_al:LIPIcs:2017:7698,
author = {Matthew de Brecht and Arno Pauly},
title = {{Noetherian Quasi-Polish spaces}},
booktitle = {26th EACSL Annual Conference on Computer Science Logic (CSL 2017)},
pages = {16:1--16:17},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-045-3},
ISSN = {1868-8969},
year = {2017},
volume = {82},
editor = {Valentin Goranko and Mads Dam},
publisher = {Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik},
address = {Dagstuhl, Germany},
URL = {http://drops.dagstuhl.de/opus/volltexte/2017/7698},
URN = {urn:nbn:de:0030-drops-76988},
doi = {10.4230/LIPIcs.CSL.2017.16},
annote = {Keywords: Descriptive set theory, synthetic topology, well-quasi orders, Noetherian spaces, compactness}
}
| Keywords: | Descriptive set theory, synthetic topology, well-quasi orders, Noetherian spaces, compactness | |
| Seminar: | 26th EACSL Annual Conference on Computer Science Logic (CSL 2017) | |
| Issue date: | 2017 | |
| Date of publication: | 16.08.2017 |
