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Apartness spaces as a framework for constructive topology

2003
*
Annals of Pure and Applied Logic
*

An axiomatic development of the theory of

doi:10.1016/s0168-0072(02)00033-7
fatcat:iisyfl3h6fhqledoi7xxzu3owi
*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. ...##
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Exact computation over topological spaces
[article]

2018
*
arXiv
*
pre-print

We retrieve existing theory and derive strong new results on the efficient representation of continuous real-valued functions defined on

arXiv:1806.01636v1
fatcat:4olgukx46ncvdeqq7jb4xgny3q
*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 ...##
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Quasi-uniformities: Reconciling domains with metric spaces
[chapter]

1988
*
Lecture Notes in Computer Science
*

We show that quasi-metric or quasi-uniform

doi:10.1007/3-540-19020-1_12
fatcat:2eg4bs7ex5herf7z7aq2ibuzbi
*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*? ...##
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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. ...##
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Pre-apartness structures on spaces of functions

2006
*
Journal of Complexity
*

Pre-

doi:10.1016/j.jco.2006.04.008
fatcat:zn4q5fi4grfndbrbd2j6yaf63y
*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. ...##
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Foreword

2006
*
Theoretical Computer Science
*

These have given rise to

doi:10.1016/j.tcs.2006.07.046
fatcat:xs4mapi5evg77kgvfqh5ushl34
*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*. ...##
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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? ...##
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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*. ...##
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Point-Set Apartness
[chapter]

2011
*
Apartness and Uniformity
*

Each point-set pre-

doi:10.1007/978-3-642-22415-7_2
fatcat:2ypa5axwgrcb3j7slejdwodr7u
*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. ...##
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A constructive theory of point-set nearness

2003
*
Theoretical Computer Science
*

An axiomatic

doi:10.1016/s0304-3975(02)00711-9
fatcat:qaow7gtupbgrvhos45dsom2pki
*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. ...##
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Natural Topology
[article]

2012
*
arXiv
*
pre-print

We give

arXiv:1210.6288v1
fatcat:vivhyx67hjaitga7keriilcl54
*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. ...##
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Precompact Apartness Spaces

2012
*
Logical Methods in Computer Science
*

The presentation lies entirely with

doi:10.2168/lmcs-8(2:15)2012
fatcat:b6xplcpeqbguzb7cs4s7uky62a
*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. ...##
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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 ...##
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Designing the Topology of Graph Neural Networks: A Novel Feature Fusion Perspective
[article]

2021
*
arXiv
*
pre-print

Then we develop

arXiv:2112.14531v1
fatcat:ozdmkssiarfo7iohewiepeiwre
*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. ...##
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Fast multiscale contrast independent preconditioners for linear elastic topology optimization problems
[article]

2020
*
arXiv
*
pre-print

The goal of this work is to present

arXiv:2006.13387v1
fatcat:lx6frcuaaffghhzlseuckwvali
*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] . ...
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