Noetherian Quasi-Polish spaces

Authors Matthew de Brecht, Arno Pauly

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Matthew de Brecht
Arno Pauly

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Matthew de Brecht and Arno Pauly. Noetherian Quasi-Polish spaces. In 26th EACSL Annual Conference on Computer Science Logic (CSL 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 82, pp. 16:1-16:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)


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.
  • Descriptive set theory
  • synthetic topology
  • well-quasi orders
  • Noetherian spaces
  • compactness


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