Wadge hardness in Scott spaces and its effectivization

Abstract

We prove some results on the Wadge order on the space of sets of natural numbers endowed with Scott topology, and more generally, on omega-continuous domains. Using alternating decreasing chains we characterize the property of Wadge hardness for the classes of the Hausdorff difference hierarchy (iterated differences of open sets). A similar characterization holds for Wadge one-to-one and finite-to-one completeness. We consider the same questions for the effectivization of the Wadge relation. We also show that for the space of sets of natural numbers endowed with the Scott topology, in each class of the Hausdorff difference hierarchy there are two strictly increasing chains of Wadge degrees of sets properly in that class. The length of these chains is the rank of the considered class, and each element in one chain is incomparable with all the elements in the other chain.

The Wadge Order on the Scott Domain Is Not a Well-Quasi-Order

“…We will restrict ourselves to the study of the quasi-order ∆ 0 2 (Pω), ≤ w . As mentioned in the Introduction, some results have already been obtained on this quasi-order in [Sel05] and [BG15]. The main result of this article (Theorem 26) comes as a generalization of a construction introduced by Selivanov in [Sel05] that we recall here.…”

“…A tree T ⊆ X < is a set of finite X -sequences closed under the prefix relation. 2 It is well founded if it has no infinite branch, 3 in which case the rank of any t ∈ T is (well) defined by -induction: rk T (t) = 0 if t is -maximal and rk T (t) = sup{rk T (s) + 1 | t s} otherwise. The rank rk(T ) of a nonempty well-founded tree T is the ordinal rk T (∅).…”

“…Several results have already been obtained by Selivanov who proved the existence of ≤ w -antichains of size 4 for P , as well as the existence of ≤ w -minimal sets at each level of the difference hierarchy of open sets [19]; and by Becher and Grigorieff who exhibited, for each infinite level α of the difference hierarchy of open sets, some strictly ≤ w -increasing chains of sets of length α, and also described the ≤ w -maximal sets for each such level for a large number of quasi-Polish spaces including P [3]. In this article, we show both that the Wadge order on the subsets of P is ill-founded and that it admits infinite antichains.…”

A Wadge hierarchy for second countable spaces

“…B Yann Pequignot yann.pequignot@liafa.univ-paris-diderot.fr; yann.pequignot@unil.ch 1 Institut des systèmes d'information, Université de Lausanne, Lausanne, Switzerland 2 LIAFA, Université Paris Diderot -Paris 7, Paris, France importance to Analysis [11], topological spaces that do not satisfy the Hausdorff separation property are central to Algebraic Geometry [6] and to Computer Science [7]. This paper considers without distinction all second countable spaces which satisfy the weakest separation property T 0 , namely every two points which have exactly the same neighbourhoods are equal.…”

“…In a more general setting, Schlicht [20] showed that in any non zero dimensional metric space there is an antichain for the qo of continuous reducibility of size continuum consisting of Borel sets. Selivanov ([21] and references there) and also Becher and Grigorieff [2] studied continuous reducibility in non Hausdorff spaces, where the situation is in general much less satisfactory than in the case of the Baire space.…”