License: Creative Commons Attribution 3.0 Unported license (CC BY 3.0)
When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.CSL.2017.16
URN: urn:nbn:de:0030-drops-76988
URL: http://dagstuhl.sunsite.rwth-aachen.de/volltexte/2017/7698/
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de Brecht, Matthew ; Pauly, Arno

Noetherian Quasi-Polish spaces

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LIPIcs-CSL-2017-16.pdf (0.6 MB)

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
Collection: 26th EACSL Annual Conference on Computer Science Logic (CSL 2017)
Issue Date: 2017
Date of publication: 16.08.2017

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