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Conference Paper/Proceeding/Abstract 10717 views 69 downloads

Noetherian Quasi-Polish spaces

Matthew de Brecht, Arno Pauly Orcid Logo

26th EACSL Annual Conference on Computer Science Logic (CSL 2017), Volume: 82

Swansea University Author: Arno Pauly Orcid Logo

DOI (Published version): 10.4230/LIPIcs.CSL.2017.16

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...

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Published in: 26th EACSL Annual Conference on Computer Science Logic (CSL 2017)
Published: Schloss Dagstuhl 2017
Online Access: http://drops.dagstuhl.de/opus/volltexte/2017/7698
URI: https://cronfa.swan.ac.uk/Record/cronfa37374
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fullrecord <?xml version="1.0"?><rfc1807><datestamp>2020-07-14T10:54:47.0194558</datestamp><bib-version>v2</bib-version><id>37374</id><entry>2017-12-08</entry><title>Noetherian Quasi-Polish spaces</title><swanseaauthors><author><sid>17a56a78ec04e7fc47b7fe18394d7245</sid><ORCID>0000-0002-0173-3295</ORCID><firstname>Arno</firstname><surname>Pauly</surname><name>Arno Pauly</name><active>true</active><ethesisStudent>false</ethesisStudent></author></swanseaauthors><date>2017-12-08</date><deptcode>SCS</deptcode><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 &#x3A3;02-subset of the space of &#x3A3;02-subsets of a given space. This suggests higher-order analogues to compactness: We can, e.g.~, investigate the spaces X where {X} is a &#x394;02-subset of the space of &#x394;02-subsets of X. Call this notion &#x2207;-compactness. As &#x394;02 is self-dual, we find that both universal and existential quantifier over &#x2207;-compact spaces preserve &#x394;02 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 &#x2207;-compact spaces: A Quasi-Polish space is Noetherian iff it is &#x2207;-compact. Note that the restriction to Quasi-Polish spaces is sufficiently general to include plenty of examples.</abstract><type>Conference Paper/Proceeding/Abstract</type><journal>26th EACSL Annual Conference on Computer Science Logic (CSL 2017)</journal><volume>82</volume><publisher>Schloss Dagstuhl</publisher><keywords>Quasi-Polish, synthetic topology, Noetherian space, finitely many mindchanges</keywords><publishedDay>31</publishedDay><publishedMonth>12</publishedMonth><publishedYear>2017</publishedYear><publishedDate>2017-12-31</publishedDate><doi>10.4230/LIPIcs.CSL.2017.16</doi><url>http://drops.dagstuhl.de/opus/volltexte/2017/7698</url><notes/><college>COLLEGE NANME</college><department>Computer Science</department><CollegeCode>COLLEGE CODE</CollegeCode><DepartmentCode>SCS</DepartmentCode><institution>Swansea University</institution><apcterm/><lastEdited>2020-07-14T10:54:47.0194558</lastEdited><Created>2017-12-08T13:10:20.8480397</Created><path><level id="1">Faculty of Science and Engineering</level><level id="2">School of Mathematics and Computer Science - Computer Science</level></path><authors><author><firstname>Matthew de</firstname><surname>Brecht</surname><order>1</order></author><author><firstname>Arno</firstname><surname>Pauly</surname><orcid>0000-0002-0173-3295</orcid><order>2</order></author></authors><documents><document><filename>0037374-08122017141945.pdf</filename><originalFilename>2017-pauly-debrecht-CSL.pdf</originalFilename><uploaded>2017-12-08T14:19:45.8770000</uploaded><type>Output</type><contentLength>588148</contentLength><contentType>application/pdf</contentType><version>Version of Record</version><cronfaStatus>true</cronfaStatus><embargoDate>2017-12-08T00:00:00.0000000</embargoDate><documentNotes>Published under a creative commons CC BY licence</documentNotes><copyrightCorrect>true</copyrightCorrect><language>eng</language></document></documents><OutputDurs/></rfc1807>
spelling 2020-07-14T10:54:47.0194558 v2 37374 2017-12-08 Noetherian Quasi-Polish spaces 17a56a78ec04e7fc47b7fe18394d7245 0000-0002-0173-3295 Arno Pauly Arno Pauly true false 2017-12-08 SCS 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 Σ02-subset of the space of Σ02-subsets of a given space. This suggests higher-order analogues to compactness: We can, e.g.~, investigate the spaces X where {X} is a Δ02-subset of the space of Δ02-subsets of X. Call this notion ∇-compactness. As Δ02 is self-dual, we find that both universal and existential quantifier over ∇-compact spaces preserve Δ02 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 ∇-compact spaces: A Quasi-Polish space is Noetherian iff it is ∇-compact. Note that the restriction to Quasi-Polish spaces is sufficiently general to include plenty of examples. Conference Paper/Proceeding/Abstract 26th EACSL Annual Conference on Computer Science Logic (CSL 2017) 82 Schloss Dagstuhl Quasi-Polish, synthetic topology, Noetherian space, finitely many mindchanges 31 12 2017 2017-12-31 10.4230/LIPIcs.CSL.2017.16 http://drops.dagstuhl.de/opus/volltexte/2017/7698 COLLEGE NANME Computer Science COLLEGE CODE SCS Swansea University 2020-07-14T10:54:47.0194558 2017-12-08T13:10:20.8480397 Faculty of Science and Engineering School of Mathematics and Computer Science - Computer Science Matthew de Brecht 1 Arno Pauly 0000-0002-0173-3295 2 0037374-08122017141945.pdf 2017-pauly-debrecht-CSL.pdf 2017-12-08T14:19:45.8770000 Output 588148 application/pdf Version of Record true 2017-12-08T00:00:00.0000000 Published under a creative commons CC BY licence true eng
title Noetherian Quasi-Polish spaces
spellingShingle Noetherian Quasi-Polish spaces
Arno Pauly
title_short Noetherian Quasi-Polish spaces
title_full Noetherian Quasi-Polish spaces
title_fullStr Noetherian Quasi-Polish spaces
title_full_unstemmed Noetherian Quasi-Polish spaces
title_sort Noetherian Quasi-Polish spaces
author_id_str_mv 17a56a78ec04e7fc47b7fe18394d7245
author_id_fullname_str_mv 17a56a78ec04e7fc47b7fe18394d7245_***_Arno Pauly
author Arno Pauly
author2 Matthew de Brecht
Arno Pauly
format Conference Paper/Proceeding/Abstract
container_title 26th EACSL Annual Conference on Computer Science Logic (CSL 2017)
container_volume 82
publishDate 2017
institution Swansea University
doi_str_mv 10.4230/LIPIcs.CSL.2017.16
publisher Schloss Dagstuhl
college_str Faculty of Science and Engineering
hierarchytype
hierarchy_top_id facultyofscienceandengineering
hierarchy_top_title Faculty of Science and Engineering
hierarchy_parent_id facultyofscienceandengineering
hierarchy_parent_title Faculty of Science and Engineering
department_str School of Mathematics and Computer Science - Computer Science{{{_:::_}}}Faculty of Science and Engineering{{{_:::_}}}School of Mathematics and Computer Science - Computer Science
url http://drops.dagstuhl.de/opus/volltexte/2017/7698
document_store_str 1
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description 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 Σ02-subset of the space of Σ02-subsets of a given space. This suggests higher-order analogues to compactness: We can, e.g.~, investigate the spaces X where {X} is a Δ02-subset of the space of Δ02-subsets of X. Call this notion ∇-compactness. As Δ02 is self-dual, we find that both universal and existential quantifier over ∇-compact spaces preserve Δ02 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 ∇-compact spaces: A Quasi-Polish space is Noetherian iff it is ∇-compact. Note that the restriction to Quasi-Polish spaces is sufficiently general to include plenty of examples.
published_date 2017-12-31T03:47:03Z
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score 11.014067

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