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A Wadge hierarchy for second countable spaces

Yann Pequignot

Archive for Mathematical Logic 54 (5):659-683 (2015)

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(1 other version)A q-wadge hierarchy in quasi-polish spaces.Victor Selivanov - 2020 - Journal of Symbolic Logic:1-26.details The wedge hierarchy was originally defined and studied only in the Baire space (and some other zero-dimensional spaces). Here we extend the Wadge hierarchy of Borel sets to arbitrary topological spaces by providing a set-theoretic definition of all its levels. We show that our extension behaves well in second countable spaces and especially in quasi-Polish spaces. In particular, all levels are preserved by continuous open surjections between second countable spaces which implies e.g. several Hausdorff-Kuratowski-type theorems in quasi-Polish spaces. In fact, (...) Download Export citation Bookmark 1 citation
(1 other version)A q-wadge hierarchy in quasi-polish spaces.Victor Selivanov - 2022 - Journal of Symbolic Logic 87 (2):732-757.details The Wadge hierarchy was originally defined and studied only in the Baire space. Here we extend the Wadge hierarchy of Borel sets to arbitrary topological spaces by providing a set-theoretic definition of all its levels. We show that our extension behaves well in second countable spaces and especially in quasi-Polish spaces. In particular, all levels are preserved by continuous open surjections between second countable spaces which implies e.g., several Hausdorff–Kuratowski -type theorems in quasi-Polish spaces. In fact, many results hold not (...) Download Export citation Bookmark 1 citation
The wadge order on the Scott domain is not a well-quasi-order.Jacques Duparc & Louis Vuilleumier - 2020 - Journal of Symbolic Logic 85 (1):300-324.details We prove that the Wadge order on the Borel subsets of the Scott domain is not a well-quasi-order, and that this feature even occurs among the sets of Borel rank at most 2. For this purpose, a specific class of countable 2-colored posets$\mathbb{P}_{emb} $equipped with the order induced by homomorphisms is embedded into the Wadge order on the$\Delta _2^0 $-degrees of the Scott domain. We then show that$\mathbb{P}_{emb} $admits both infinite strictly decreasing chains and infinite antichains with respect to this (...) Download Export citation Bookmark
Continuous reducibility and dimension of metric spaces.Philipp Schlicht - 2018 - Archive for Mathematical Logic 57 (3-4):329-359.details If is a Polish metric space of dimension 0, then by Wadge’s lemma, no more than two Borel subsets of X are incomparable with respect to continuous reducibility. In contrast, our main result shows that for any metric space of positive dimension, there are uncountably many Borel subsets of that are pairwise incomparable with respect to continuous reducibility. In general, the reducibility that is given by the collection of continuous functions on a topological space \\) is called the Wadge quasi-order (...) Download Export citation Bookmark 2 citations
A syntactic approach to Borel functions: some extensions of Louveau’s theorem.Takayuki Kihara & Kenta Sasaki - 2023 - Archive for Mathematical Logic 62 (7):1041-1082.details Louveau showed that if a Borel set in a Polish space happens to be in a Borel Wadge class $$\Gamma $$, then its $$\Gamma $$ -code can be obtained from its Borel code in a hyperarithmetical manner. We extend Louveau’s theorem to Borel functions: If a Borel function on a Polish space happens to be a $$ \underset{\widetilde{}}{\varvec{\Sigma }}\hbox {}_t$$ -function, then one can find its $$ \underset{\widetilde{}}{\varvec{\Sigma }}\hbox {}_t$$ -code hyperarithmetically relative to its Borel code. More generally, we prove (...) Download Export citation Bookmark

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