We construct the space of valuations on a quasi-Polish space in terms of the characterization of quasi-Polish spaces as spaces of ideals of a countable transitive relation. Our construction is closely related to domain theoretical work on the probabilistic powerdomain, and helps illustrate the connections between domain theory and quasi-Polish spaces. Our approach is consistent with previous work on computable measures, and can be formalized within weak formal systems, such as subsystems of second order arithmetic.

We investigate powerspace constructions on topological spaces, with a particular focus on the category of quasi-Polish spaces. We show that the upper and lower powerspaces commute on all quasi-Polish spaces, and show more generally that this commutativity is equivalent to the topological property of consonance. We then investigate powerspace constructions on the open set lattices of quasi-Polish spaces, and provide a complete characterization of how the upper and lower powerspaces distribute over the open set lattice construction.

We identify four countable topological spaces $S_2$, $S_1$, $S_D$, and $S_0$ which serve as canonical examples of topological spaces which fail to be quasi-Polish. These four spaces respectively correspond to the $T_2$, $T_1$, $T_D$, and $T_0$-separation axioms. $S_2$ is the space of rationals, $S_1$ is the natural numbers with the cofinite topology, $S_D$ is an infinite chain without a top element, and $S_0$ is the set of finite sequences of natural numbers with the lower topology induced by the prefix ordering. Our main result is a generalization of Hurewicz's theorem showing that a co-analytic subset of a quasi-Polish space is either quasi-Polish or else contains a countable $\Pi^0_2$-subset homeomorphic to one of these four spaces.

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 $Σ^0_2$-subset of the space of $Σ^0_2$-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 $Δ^0_2$-subset of the space of $Δ^0_2$-subsets of $\mathbf{X}$. Call this notion $\nabla$-compactness. As $Δ^0_2$ is self-dual, we find that both universal and existential quantifier over $\nabla$-compact spaces preserve $Δ^0_2$ 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.

We propose to extend descriptive set theory (DST) beyond its traditional setting of Polish spaces to the represented spaces. There, we can reformulate DST in terms of endofunctors on the categories of represented spaces and computable or continuous functions. In particular, this approach satisfies the demand for a uniform approach to both classic and effective DST -- computability follows naturally from the setting, rather than having to be explicitly demanded. The previous endeavour to extend DST to the Quasi-Polish spaces is subsumed by this work. In several cases the category-theoretic setting enables new, very succinct proofs, and sheds a new light on why certain results are true. The framework lets us make formal some natural questions not easily approachable by traditional methods.