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Wadge hardness in Scott spaces and its effectivization

Published online by Cambridge University Press:  14 November 2014

VERÓNICA BECHER  and
SERGE GRIGORIEFF
VERÓNICA BECHER
Affiliation:
FCEyN, Universidad de Buenos Aires & CONICET, Argentina Email: [email protected]
SERGE GRIGORIEFF
Affiliation:
LIAFA, CNRS & Université Paris Diderot - Paris 7, France Email: [email protected]
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Abstract

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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.

Type
Paper
Information
Mathematical Structures in Computer Science , Volume 25 , Special Issue 7: Computing with Infinite Data: Topological and Logical Foundations Part 1 , October 2015 , pp. 1520 - 1545
Copyright
Copyright © Cambridge University Press 2014 

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