Noetherian Quasi-Polish Spaces

Abstract

In the presence of suitable power spaces, compactness of $\mathbf{X}$ can be characterized as the singleton $\{X\}$ being open in the space $\mathcal{O}(\mathbf{X})$ of open subsets of $\mathbf{X}$. 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 $\mathbf{X}$ where $\{X\}$ is a $\Delta^0_2$-subset of the space of $\Delta^0_2$-subsets of $\mathbf{X}$. 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.