par De Brecht, Matthew;Pauly, Arno
Référence Leibniz international proceedings in informatics, 82, 16
Publication Publié, 2017-08
Article révisé par les pairs
Résumé : 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 r-compactness. As Δ02 is self-dual, we find that both universal and existential quantifier over r-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 r-compact spaces: A Quasi-Polish space is Noetherian iff it is r-compact. Note that the restriction to Quasi-Polish spaces is sufficiently general to include plenty of examples.