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A Q-WADGE HIERARCHY IN QUASI-POLISH SPACES

Part of: Partial differential equations, initial value and time-dependent initial-boundary value problems Partial differential equations on manifolds; differential operators Computability and recursion theory

Published online by Cambridge University Press:  05 October 2020

VICTOR SELIVANOV
VICTOR SELIVANOV*
Affiliation:
A.P. ERSHOV INSTITUTE OF INFORMATICS SYSTEMS SB RAS AC. LAVRENTYEV AVE. 6, NOVOSIBIRSK, 630090, RUSSIAE-mail: vseliv@iis.nsk.su
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Abstract

The Wadge 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 (HK)-type theorems in quasi-Polish spaces. In fact, many results hold not only for the Wadge hierarchy of sets but also for its extension to Borel functions from a space to a countable better quasiorder Q.

Keywords

Borel hierarchyWadge hierarchyfine hierarchyiterated labeled treeh-quasiorderbetter quasiorderQ-partition

MSC classification

Primary: 03D15: Complexity of computation (including implicit computational complexity)
Secondary: 03D78: Computation over the reals 58J45: Hyperbolic equations 65M06: Finite difference methods 65M25: Method of characteristics
Type
Article
Information
The Journal of Symbolic Logic , Volume 87 , Issue 2 , June 2022 , pp. 732 - 757
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Copyright
© The Author(s), 2020. Published by Cambridge University Press on behalf of The Association for Symbolic Logic

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