Noetherian Quasi-Polish Spaces

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 Sigma02-subset of the space of Sigma02-subsets of a given space. This suggests higher-order analogues to compactness: We can, e.g.~, investigate the spaces X where X is a Delta02-subset of the space of Delta02-subsets of X. Call this notion nabla-compactness. As Delta02 is self-dual, we find that both universal and existential quantifier over nabla-compact spaces preserve Delta02 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.