14 research outputs found
A generalization of a theorem of Hurewicz for quasi-Polish spaces
We identify four countable topological spaces , , , and
which serve as canonical examples of topological spaces which fail to be
quasi-Polish. These four spaces respectively correspond to the , ,
, and -separation axioms. is the space of rationals, is
the natural numbers with the cofinite topology, is an infinite chain
without a top element, and 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
-subset homeomorphic to one of these four spaces
Computability on quasi-Polish spaces
International audienceWe investigate the effectivizations of several equivalent definitions of quasi-Polish spaces and study which characterizations hold effectively. Being a computable effectively open image of the Baire space is a robust notion that admits several characterizations. We show that some natural effectivizations of quasi-metric spaces are strictly stronger
Overt choice
We introduce and study the notion of overt choice for countably-based spaces and for CoPolish spaces. Overt choice is the task of producing a point in a closed set specified by what open sets intersect it. We show that the question of whether overt choice is continuous for a given space is related to topological completeness notions such as the Choquet-property; and to whether variants of Michaelâs selection theorem hold for that space. For spaces where overt choice is discontinuous it is interesting to explore the resulting Weihrauch degrees, which in turn are related to whether or not the space is FrĂ©chetâUrysohn
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 ÎŁ02-subset of the space of ÎŁ02-subsets of a given space. This suggests higher-order analogues to compactness: We can, e.g.~, investigate the spaces X where {X} is a Î02-subset of the space of Î02-subsets of X. Call this notion â-compactness. As Î02 is self-dual, we find that both universal and existential quantifier over â-compact spaces preserve Î02 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 â-compact spaces: A Quasi-Polish space is Noetherian iff it is â-compact. Note that the restriction to Quasi-Polish spaces is sufficiently general to include plenty of examples