Noetherian Quasi-Polish spaces - Arno Pauly (26th EACSL Annual Conference on Computer Science Logic (CSL 2017), 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 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.