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Apartness spaces as a framework for constructive topology

Douglas Bridges, Luminiţa Vîţă
2003 Annals of Pure and Applied Logic  
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]

Frank Waaldijk
2018 arXiv   pre-print
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]

M. B. Smyth
1988 Lecture Notes in Computer Science  
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  
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

D.S. Bridges, L.S. Vıˆţă
2006 Journal of Complexity  
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

Ralph Kopperman, Prakash Panangaden, Michael B. Smyth, Dieter Spreen, Julian Webster
2006 Theoretical Computer Science  
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  
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  
; 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]

Douglas S. Bridges, Luminiţa Simona Vîţă
2011 Apartness and Uniformity  
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

L.S. Vîţă, D.S. Bridges
2003 Theoretical Computer Science  
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]

Frank Waaldijk
2012 arXiv   pre-print
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

Douglas Bridges, Martín Escardó
2012 Logical Methods in Computer Science  
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  
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]

Lanning Wei, Huan Zhao, Zhiqiang He
2021 arXiv   pre-print
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]

Miguel Zambrano, Sintya Serrano, Boyan S. Lazarov, Juan Galvis
2020 arXiv   pre-print
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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