Quasi-Polish spaces

Abstract

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 Σ 0 2subset 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 X where {X} is a ∆ 0 2 -subset of the space of ∆ 0 2 -subsets of X. Call this notion ∇-compactness. As ∆ 0 2 is self-dual, we find that both universal and existential quantifier over ∇-compact spaces preserve ∆ 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 ∇-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.

Total Representations

“…Note that for Polish spaces the class Σ 1 1 of analytic sets has several nice equivalent characterizations (in particular, as the class of continuous images of Polish spaces or as the class of sets obtained from the closed sets by applying the Suslin A-operation). In Lemma 56 of [Bre13] it was observed that the characterization in terms of continuous images extends to the quasi-Polish spaces. In Section 8 we will see that this characterization fails in general for non-countably based admissibly totally represented spaces.…”

“…In [Bre13] the following important characterization of quasi-Polish spaces in terms of Borel hierarchy was obtained. Proposition 3.6.…”

“…A space X is Polish if it is countably based and metrizable with a metric d such that (X, d) is a complete metric space. A space X is quasi-Polish [Bre13] if it is countably based and quasi-metrizable with a quasi-metric d such that (X, d) is a complete quasi-metric space. A quasi-metric on X is a function from X × X to the nonnegative reals such that d(x, y) = d(y, x) = 0 iff x = y, and d(x, y) ≤ d(x, z) + d(z, y).…”

“…A quasi-metric on X is a function from X × X to the nonnegative reals such that d(x, y) = d(y, x) = 0 iff x = y, and d(x, y) ≤ d(x, z) + d(z, y). Since for the quasimetric spaces different notions of completeness and of a Cauchy sequence are considered, the definition of quasi-Polish spaces should be made more precise (see [Bre13] for additional details). We skip these details because we will in fact use other characterizations of these spaces given in the sequel.…”

“…Although the analogy of representation theory to numbering theory is well known, there is a striking difference between them: numberings are in most cases total functions while representations considered so far are almost always partial functions (among rare exceptions are Section 1.1.6 of [Ba00] and [Bre13]). Note that total numberings have a better theory than partial numberings and are sufficient for many important topics like computable model theory (although partial numberings also have some advantages, in particular the corresponding category is known to be cartesian closed while the category of total numberings is not).…”