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Apartness spaces as a framework for constructive topology
2003
Annals of Pure and Applied Logic
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An axiomatic development of the theory of apartness and nearness of a point and a set is introduced as a framework for constructive topology. ...
Various notions of continuity of mappings between apartness spaces are compared; the constructive independence of one of the axioms from the others is demonstrated; and the product apartness structure ...
Acknowledgement: The authors thank Peter Schuster for his numerous helpful comments on drafts of this paper, and for his continuing support and advice. ...
doi:10.1016/s0168-0072(02)00033-7
fatcat:iisyfl3h6fhqledoi7xxzu3owi
Exact computation over topological spaces
[article]
2018
arXiv
pre-print
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We retrieve existing theory and derive strong new results on the efficient representation of continuous real-valued functions defined on a general class of topological spaces (called natural spaces). ...
We give an exposition of Natural Topology (NToP), which highlights its advantages for exact computation. ...
THEORY AND PRACTICE IN CONSTRUCTIVE TOPOLOGY 1.0 A suitable framework for constructive topology In recent decades, the problem of finding a suitable framework for constructive topology has received increasing ...
arXiv:1806.01636v1
fatcat:4olgukx46ncvdeqq7jb4xgny3q
Quasi-uniformities: Reconciling domains with metric spaces
[chapter]
1988
Lecture Notes in Computer Science
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We show that quasi-metric or quasi-uniform spaces provide, inter alia, a common generalization of cpo's and metric spaces as used in denotational semantics. ...
Specific results include general fixed point theorem and a sequential completion construction. ...
II for definitions). can be trivially represented as a quasi-metric space by Any cpo (D,_C) putting 0 if xC--y d (x ,y) = 1 otherwise But what about the topology? ...
doi:10.1007/3-540-19020-1_12
fatcat:2eg4bs7ex5herf7z7aq2ibuzbi
Page 9557 of Mathematical Reviews Vol. , Issue 2004m
[page]
2004
Mathematical Reviews
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Jpn. 56 (2002), no. 1, 123 132; MR 2004b:54043] as the definitive one for our apartness theory. ...
As an example of formal topology, the continuum is presented as a formal space. Then the basic picture is introduced. This is a formalisation of the relations between the formal opens and points. ...
Pre-apartness structures on spaces of functions
2006
Journal of Complexity
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Pre-apartness structures are defined on Y X , where X is an inhabited set and Y a uniform space. ...
These structures clarify the discussion of proximal and uniform convergence in the constructive theory of apartness spaces. ...
Acknowledgment The authors thank the New Zealand Foundation for Science & Technology for supporting Luminiţa Vîţȃ as a Postdoctoral Research Fellow from 2000 to 2005. ...
doi:10.1016/j.jco.2006.04.008
fatcat:zn4q5fi4grfndbrbd2j6yaf63y
Foreword
2006
Theoretical Computer Science
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These have given rise to a constructive treatment of topological spaces, Formal Topology. Constructive theories have the important property that algorithms can be derived from their proofs. ...
as a partial metric or a measurement. ...
them as certain closure spaces. ...
doi:10.1016/j.tcs.2006.07.046
fatcat:xs4mapi5evg77kgvfqh5ushl34
Page 7149 of Mathematical Reviews Vol. , Issue 2004i
[page]
2004
Mathematical Reviews
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Summary: “An axiomatic constructive development of the theory of nearness and apartness of a point and a set is introduced as a setting for topology.”
20041:54032 54E15 54A20
Freire, Nuno C. ...
Summary: “In this paper we describe a construction of a large class of hyperconvex metric spaces. In particular, this construction contains well-known examples of hyperconvex spaces such as R? ...
Page 6939 of Mathematical Reviews Vol. , Issue 2003i
[page]
2003
Mathematical Reviews
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; Christchurch) Apartness spaces as a framework for constructive topology. ...
The authors introduce an axiomatic development of the theory of apartness and nearness of a point and a set as a framework for constructive topology. ...
Point-Set Apartness
[chapter]
2011
Apartness and Uniformity
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Each point-set pre-apartness gives rise to a topology-the apartness topology-on X, and to several constructively distinct continuity properties, which are explored in Section 2.3. ...
Synopsis We first introduce the notion of a (pre-)apartness between points and subsets in an abstract space X, and derive some elementary properties from our axioms. ...
It is left as an exercise for the reader to show that this topology has the reverse Kolmogorov property (2.14). To establish A5 for the corresponding topological pre-apartness , we argue as follows. ...
doi:10.1007/978-3-642-22415-7_2
fatcat:2ypa5axwgrcb3j7slejdwodr7u
A constructive theory of point-set nearness
2003
Theoretical Computer Science
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An axiomatic constructive development of the theory of nearness and apartness of a point and a set is introduced as a setting for topology. t a), d.bridges@math.canterbury.ac.nz (D.S. ...
Bridges). 1 This is not to say, or even suggest, that we are uninterested in philosophical constructivism; see, for example, [9] . ...
They also thank the Marsden Fund of the Royal Society of New Zealand for supporting Luminit a Vˆ t a's research from 1997 on. ...
doi:10.1016/s0304-3975(02)00711-9
fatcat:qaow7gtupbgrvhos45dsom2pki
Natural Topology
[article]
2012
arXiv
pre-print
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We give a theoretical and applicable framework for dealing with real-world phenomena. ...
We study quotients of Baire space, and obtain constructive metrizability of star-finitary spaces. Silva spaces arise as example of non-metrizable natural spaces. ...
Remember that for a spraid (V , T # ) derived from (V, #, ), and subsets A, B of V we write A #B iff #b for all ∈ A, b ∈ B. We write A ≈ B iff ≈ b for some ∈ A, b ∈ B. ...
arXiv:1210.6288v1
fatcat:vivhyx67hjaitga7keriilcl54
Precompact Apartness Spaces
2012
Logical Methods in Computer Science
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The presentation lies entirely with a Bishop-style constructive framework, and is a contribution to the ongoing development of the constructive theories of apartness and uniformity. ...
We present a notion of precompactness, and study some of its properties, in the context of apartness spaces whose apartness structure is not necessarily induced by any uniform one. ...
This work was done in part when the author was a guest of Professor Helmut Schwichtenberg in the Logic Section of the Mathematisches Institut, Ludwig-Maximilians-Universität, München. ...
doi:10.2168/lmcs-8(2:15)2012
fatcat:b6xplcpeqbguzb7cs4s7uky62a
Page 4598 of Mathematical Reviews Vol. , Issue 94h
[page]
1994
Mathematical Reviews
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The paper under review provides some nontrivial constructions for generating circuits in n-space. ...
The graph G is said to be a (generic) circuit in n-space if the deletion of any edge does not decrease the degrees of freedom of any realization as a bar framework, but the deletion of any two such edges ...
Designing the Topology of Graph Neural Networks: A Novel Feature Fusion Perspective
[article]
2021
arXiv
pre-print
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2022 accepted doi:10.1145/3485447.3512185 arXiv:2112.14531v2 fatcat:wz23dnnyjjedrjzfiu4mtogcey
Then we develop a neural architecture search method on top of the unified framework which contains a set of selection and fusion operations in the search space and an improved differentiable search algorithm ...
To be specific, we provide a feature fusion perspective in designing GNN topology and propose a novel framework to unify the existing topology designs with feature selection and fusion strategies. ...
We also thank all anonymous reviewers for their constructive comments, which help us to improve the quality of this manuscript. ...
arXiv:2112.14531v1
fatcat:ozdmkssiarfo7iohewiepeiwre
Fast multiscale contrast independent preconditioners for linear elastic topology optimization problems
[article]
2020
arXiv
pre-print
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The goal of this work is to present a fast and viable approach for the numerical solution of the high-contrast state problems arising in topology optimization. ...
The solvers are based on two-levels domain decomposition techniques with a carefully constructed coarse level to deal with the high-contrast and multi-scale nature of the problem. ...
That is precisely the case for topology optimized linear elastic structures. These design features can be observed in Figure 1 as well as in previous articles on the topic [20, 21, 22] . ...
arXiv:2006.13387v1
fatcat:lx6frcuaaffghhzlseuckwvali
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