de Brecht, Matthew ;
Pauly, Arno
Noetherian QuasiPolish spaces
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_2subset of the space of Sigma^0_2subsets of a given space. This suggests higherorder analogues to compactness: We can, e.g., investigate the spaces X where {X} is a Delta^0_2subset of the space of Delta^0_2subsets of X. Call this notion nablacompactness. As Delta^0_2 is selfdual, we find that both universal and existential quantifier over nablacompact spaces preserve Delta^0_2 predicates.
Recall that a space is called Noetherian iff every subset is compact. Within the setting of QuasiPolish spaces, we can fully characterize the nablacompact spaces: A QuasiPolish space is Noetherian iff it is nablacompact. Note that the restriction to QuasiPolish 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 QuasiPolish spaces}},
booktitle = {26th EACSL Annual Conference on Computer Science Logic (CSL 2017)},
pages = {16:116:17},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {9783959770453},
ISSN = {18688969},
year = {2017},
volume = {82},
editor = {Valentin Goranko and Mads Dam},
publisher = {Schloss DagstuhlLeibnizZentrum fuer Informatik},
address = {Dagstuhl, Germany},
URL = {http://drops.dagstuhl.de/opus/volltexte/2017/7698},
URN = {urn:nbn:de:0030drops76988},
doi = {10.4230/LIPIcs.CSL.2017.16},
annote = {Keywords: Descriptive set theory, synthetic topology, wellquasi orders, Noetherian spaces, compactness}
}
16.08.2017
Keywords: 

Descriptive set theory, synthetic topology, wellquasi orders, Noetherian spaces, compactness 
Seminar: 

26th EACSL Annual Conference on Computer Science Logic (CSL 2017)

Issue date: 

2017 
Date of publication: 

16.08.2017 