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On the Difference Hierarchy in Countably Based T0-Spaces

2008
*
Electronical Notes in Theoretical Computer Science
*

We establish some results

doi:10.1016/j.entcs.2008.12.022
fatcat:nhetciwsl5fo5erktj35jlavu4
*on*some variants of*the*Hausdorff*difference**hierarchy*. ... We discuss also a broad class of effective topological*spaces*closely relevant to our study of*the**difference**hierarchy*and to computability*in*topology. ...*Difference**Hierarchies*of Sets*In*this section we improve some previous results*on**the*DH,*in*particular extend*the*theory*in*[19] to*the*ω-continuous domains. ...##
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Type Classes and Filters for Mathematical Analysis in Isabelle/HOL
[chapter]

2013
*
Lecture Notes in Computer Science
*

*The*improvements are centered

*on*two primary contributions: a generic theory of limits

*based*

*on*filters, and a new

*hierarchy*of type classes that includes various topological, metric, vector, and algebraic ... These let us apply many results

*in*multivariate analysis to types which are not Euclidean

*spaces*, such as

*the*extended real numbers, bounded continuous functions, or finite maps. ... Acknowledgements We want to thank John Harrison and his colleagues for

*the*development of HOL Light's multivariate analysis. ...

##
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Towards a descriptive set theory for domain-like structures

2006
*
Theoretical Computer Science
*

This is a survey of results

doi:10.1016/j.tcs.2006.07.053
fatcat:zw4tsdo5ozahfpeiyuelgg4mom
*in*descriptive set theory for domains and similar*spaces*, with*the*emphasis*on**the*-algebraic domains. ... Since*the*subject is still*in*its beginning, we discuss*in*detail several open questions and possible future development. We also mention some relevant facts of (effective) descriptive set theory. ... I am grateful to*the*university for supporting my stays and to Dieter Spreen for making*the*stays possible. I thank also an anonymous referee for*the*careful reading and useful remarks. ...##
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Total Representations
[article]

2013
*
arXiv
*
pre-print

Almost all representations considered

arXiv:1304.1239v1
fatcat:3kp275ziknfqfamsvzduxvvwkm
*in*computable analysis are partial. We provide arguments*in*favor of total representations (by elements of*the*Baire*space*). ... of topological*spaces*relevant to computable analysis. ... I also thank Vasco Brattka, Peter Hertling and Matthias Schröder for stimulating discussions, and to*the*anonymous referees for many useful suggestions and bibliographical hints. ...##
###
MEASURE-THEORETIC UNIFORMITY IN RECURSION THEORY AND SET THEORY
[chapter]

1999
*
World Scientific Series in 20th Century Mathematics
*

*The*recursion-theoretic concepts touched

*on*are :

*the*arithmetic, hyperarithmetic and analytic

*hierarchies*of Kleene [12]; co-models of

*the*hyperarithmetic comprehension axiom of Kreisel [16]; recursive ... He shows: if ZF (Zermelo-Fraenkel set theory) is consistent, then ZF+ "there exists a translationinvariant,

*countably*additive extension of Lebesque measure defined

*on*all sets of reals"+ "

*the*dependent ... Think of (2N)% as

*the*product of

*countably*many copies of 2N and finitely many

*one*-point

*spaces*,

*one*

*one*-point

*space*{At} for each i e K. ...

##
###
Measure-Theoretic Uniformity in Recursion Theory and Set Theory

1969
*
Transactions of the American Mathematical Society
*

*The*recursion-theoretic concepts touched

*on*are :

*the*arithmetic, hyperarithmetic and analytic

*hierarchies*of Kleene [12]; co-models of

*the*hyperarithmetic comprehension axiom of Kreisel [16]; recursive ... He shows: if ZF (Zermelo-Fraenkel set theory) is consistent, then ZF+ "there exists a translationinvariant,

*countably*additive extension of Lebesque measure defined

*on*all sets of reals"+ "

*the*dependent ... Think of (2N)% as

*the*product of

*countably*many copies of 2N and finitely many

*one*-point

*spaces*,

*one*

*one*-point

*space*{At} for each i e K. ...

##
###
Measure-theoretic uniformity in recursion theory and set theory

1969
*
Transactions of the American Mathematical Society
*

*The*recursion-theoretic concepts touched

*on*are :

*the*arithmetic, hyperarithmetic and analytic

*hierarchies*of Kleene [12]; co-models of

*the*hyperarithmetic comprehension axiom of Kreisel [16]; recursive ... He shows: if ZF (Zermelo-Fraenkel set theory) is consistent, then ZF+ "there exists a translationinvariant,

*countably*additive extension of Lebesque measure defined

*on*all sets of reals"+ "

*the*dependent ... Think of (2N)% as

*the*product of

*countably*many copies of 2N and finitely many

*one*-point

*spaces*,

*one*

*one*-point

*space*{At} for each i e K. ...

##
###
Topological Ramsey Spaces Dense in Forcings
[article]

2018
*
arXiv
*
pre-print

*In*cases where a forcing turned out to be equivalent to a topological Ramsey

*space*,

*the*strong Ramsey-theoretic techniques have aided

*in*a fine-tuned analysis of

*the*Rudin-Keisler and Tukey structures ... Each topological Ramsey

*space*is endowed with a partial ordering which can be modified to a σ-closed 'almost reduction' relation analogously to

*the*partial ordering of 'mod finite'

*on*[ω]^ω. ... An ultrafilter U

*on*a

*countable*

*base*set B is a filter such that each subset of B or its complement is

*in*U. ...

##
###
RNA Shape Space Topology

2000
*
Artificial Life
*

*one*phenotype is accessible from another through changes at

*the*genotypic level. ...

*The*shape

*space*of RNA secondary structures is used to illustrate this approach. We show that evolutionary trajectories are continuous if and only if they follow connected paths

*in*phenotype

*space*. ... This article was supported

*in*part by Austrian Fonds zur Förderung der Wissenschaftlichen Forschung Proj. No. P13093-GEN. This paper is LA-UR 99:1324. ...

##
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On the complexity of Borel equivalence relations with some countability property
[article]

2018
*
arXiv
*
pre-print

*In*order to do this, we prove some extensions of

*the*Louveau-Saint Raymond theorem which itself generalized

*the*Hurewicz theorem characterizing when a Borel subset of a Polish

*space*is G δ . ...

*In*particular , we characterize when a Borel equivalence relation with

*countable*equivalence classes is Σ 0 ξ (or Π 0 ξ). ...

*The*most classical

*hierarchy*of topological complexity

*in*descriptive set theory is

*the*

*one*given by

*the*Borel classes. ...

##
###
Hierarchies of effective descriptive set theory

1969
*
Transactions of the American Mathematical Society
*

*The*roots of modern

*hierarchy*theory lie

*in*

*the*investigations of Baire, Borel, Lebesgue, and others around

*the*turn of

*the*century. ... Since each Borel set can be constructed by iteration of

*the*elementary operations of

*countable*union and complementation over a

*countable*well-ordering, it was

*in*general agreed that these sets are constructively ...

*In*this section we give a second characterization of sets of

*the*form R(XsAP(s))(F)

*in*terms of a set derivative

*based*

*on*P. ...

##
###
Hierarchies of Effective Descriptive Set Theory

1969
*
Transactions of the American Mathematical Society
*

*The*roots of modern

*hierarchy*theory lie

*in*

*the*investigations of Baire, Borel, Lebesgue, and others around

*the*turn of

*the*century. ... Since each Borel set can be constructed by iteration of

*the*elementary operations of

*countable*union and complementation over a

*countable*well-ordering, it was

*in*general agreed that these sets are constructively ...

*In*this section we give a second characterization of sets of

*the*form R(XsAP(s))(F)

*in*terms of a set derivative

*based*

*on*P. ...

##
###
Concepts of quantum non-Markovianity: A hierarchy

2018
*
Physics reports
*

*In*this report we compare

*the*descriptions of classical stochastic processes and quantum stochastic processes (as arising

*in*OQSs), and show that there are inherent

*differences*that lead to

*the*non-trivial ...

*In*fact, quantum non-Markovianity is highly context-dependent.

*The*results

*in*this report, summarized as a

*hierarchy*figure, bring clarity to

*the*nature of quantum non-Markovianity. ... Acknowledgements 58 Acknowledgements

*On*

*the*connection between dynamical decoupling and quantum Markovianity, we acknowledge motivating discussions with Daniel Burgarth and Lorenza Viola. ...

##
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Definable Operations On Weakly Recognizable Sets of Trees

2011
*
Foundations of Software Technology and Theoretical Computer Science
*

*In*order to prevent this, alternation needs to be restricted to

*the*choice of direction

*in*

*the*tree. ... Alternating automata

*on*infinite trees induce operations

*on*languages which do not preserve natural equivalence relations, like having

*the*same Mostowski-Rabin index,

*the*same Borel rank, or being continuously ... Acknowledgements

*The*second author is supported by a grant from

*the*SNFS, n. PBLAP2-132006, while

*the*third author is supported by

*the*Polish government grant no. N N206 567840. ...

##
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Normal Numbers and the Borel Hierarchy
[article]

2013
*
arXiv
*
pre-print

Similarly,

arXiv:1311.0335v1
fatcat:4vu3ksqrhjgz3afnhvefynnlca
*the*set of absolutely normal numbers is Π^0_3-complete*in**the*effective Borel*hierarchy*. ... We show that*the*set of absolutely normal numbers is Π^0_3-complete*in**the*Borel*hierarchy*of subsets of real numbers. ... This research received support from CONICET and Agencia Nacional de Promoción Científica y Tecnológica, Argentina, from*the*National Science ...
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