Noetherian Quasi-Polish spaces

In the presence of suitable power spaces, compactness ofX can be characterized as the singleton {∅} being open in the space A(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 Σ 2 subset of the space of Σ 2 -subsets of a given space. This suggests higher-order analogues to compactness: We can, e.g. , investigate the spaces X where {∅} is a ∆ 2 -subset of the space of ∆ 2 -subsets of X. Call this notion ∇-compactness. As ∆ 2 is self-dual, we find that both universal and existential quantifier over ∇-compact spaces preserve ∆ 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 ∇-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.