DOCSLIB.ORG

Approach Merotopological Spaces and Their Completion

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Approach Merotopological Spaces and Their Completion Hindawi Publishing Corporation International Journal of Mathematics and Mathematical Sciences Volume 2010, Article ID 409804, 16 pages doi:10.1155/2010/409804 Research Article Approach Merotopological Spaces and their Completion Mona Khare and Surabhi Tiwari Department of Mathematics, University of Allahabad, Allahabad 211002, India Correspondence should be addressed to Mona Khare, [email protected] Received 24 July 2009; Accepted 13 April 2010 Academic Editor: Richard Wilson Copyright q 2010 M. Khare and S. Tiwari. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. This paper introduces the concept of an approach merotopological space and studies its category- theoretic properties. Various topological categories are shown to be embedded into the category whose objects are approach merotopological spaces. The order structure of the family of all approach merotopologies on a nonempty set is discussed. Employing the theory of bunches, bunch completion of an approach merotopological space is constructed. The present study is a unified look at the completion of metric spaces, approach spaces, nearness spaces, merotopological spaces, and approach merotopological spaces. 1. Introduction Some of the applications of nearness-like structures within topology are unification, extensions, homology, and connectedness. The categories of R0-topological spaces, uniform spaces 1, 2, proximity spaces 2, 3, and contiguity spaces 4, 5 are embedded into the category of nearness spaces. The study of proximity, contiguity, and merotopic spaces in the more generalized setting of L-fuzzy theory can be seen in 6–13.In14, the notion of an approach space was introduced via different equivalent set of axioms to measure the degree of nearness between a set and a point. While developing the theory of approach spaces, Lowen et al. many a time employed tools from nearness-like structures. The notion of “distance” in approach spaces is closely related to the notion of nearness; further proximity and nearness concepts arise naturally in the context of approach spaces as can be seen in 15– 18. Hence it became mandatory looking into the nearness-like concepts in approach theory, more clearly. With the same spirit, Lowen and Lee 19 made an attempt to measure how near a collection of sets is and, in the process, axiomatized the two equivalent concepts: approach merotopic and approach seminearness structures, respectively, to measure the degree of 2 International Journal of Mathematics and Mathematical Sciences smallness and nearness of an arbitrary collection of sets, and therefore generalized approach spaces in a sense. In 2004, Bentley and Herrlich 20 gave the idea of merotopological spaces as a supercategory of many of the above mentioned categories. They also constructed the functorial completion of merotopological spaces employing the theory of bunches in merotopological spaces. In 21, we axiomatized the notion of approach nearness by adding to the axioms of an approach merotopy the axiom relating a collection of sets and the closure induced by the respective approach merotopy; and we analogously obtained cluster completion of an approach nearness space. Prerequisites for the paper are collected in Section 2.InSection 3, we axiomatize approach merotopological spaces. The category AMT of approach merotopological spaces and their respective morphisms is shown to be a topological construct and a supercategory of some of the known topological categories, including the category of topological spaces and continuous maps. Order structure of the family of all approach merotopologies on X is also discussed. In Section 4, bunch completion of an approach merotopological space is constructed, employing the theory of bunches. The concept of regularity in an approach merotopic space is introduced to obtain a relationship between cluster and bunch completion of an approach nearness space. Indeed, it is shown that cluster completion is a retract of the bunch completion of a regular approach nearness space X, ν. 2. Preliminaries and Basic Results Let X be a nonempty ordinary set. The power set of X is denoted by PX and the family of P P2 ℵ all subsets of X is denoted by X . We denote by 0 the first infinite cardinal number, by |A| the cardinality of A where A ⊆ X, and by J an arbitrary index set. For A, B subsets of PX, A∨B≡{A ∪ B : A ∈A,B ∈B}; A corefines B written as A≺B if and only if for all A ∈Athere exists B ∈Bsuch that B ⊆ A. For A⊆PX, stackA{A ⊆ X : B ⊆ A for some B ∈A}and sec A {B ⊆ X : A ∩ B / ∅, for all A ∈A} {B ⊆ X : X − B /∈ stackA}. Observe that sec2A stack A, for all A∈P2X. A grill on X is a subset G of PX satisfying ∅ /∈G; if A ∈Gand A ⊆ B, then B ∈G;andifA ∪ B ∈G, then A ∈Gor B ∈G. For basic definitions and results of merotopic spaces and nearness spaces, we refer to 1. Definition 2.1 see 20.Amerotopological space is the triple X, ξ, cl, where ξ is a merotopy and cl is a Kuratowski closure operator on X such that {clA : A ∈A}∈ξ ⇒A∈ξ, for all A∈P2X. Definition 2.2 see 19, 21.Afunctionν : P2X → 0, ∞ is called an approach merotopy on X if for any A, B∈P2X the following conditions are satisfied: AM1 A≺B⇒νA ≤ νB, AM2 A / ∅⇒νA0, AM3 ∅∈A⇒νA∞, AM4 νA∨B ≥ νA ∧ νB. The pair X, ν is called an approach merotopic space. For an approach merotopic space X, ν, { ∈ {{ } } } ⊆ we define clν A x X : ν x ,A 0 , for all A X. Then clν is a Cechˇ closure operator on X. International Journal of Mathematics and Mathematical Sciences 3 An approach merotopy ν on X is called an approach nearness on X 21 if the following condition is satisfied: { ∈A} ≥ A AN5 ν clν A : A ν . A{ ∈A} In this case, clν is a Kuratowski closure operator on X. Denote cl cl A : A . Definition 2.3 see 21.LetA∈P2X and ν be an approach merotopy on X. Then we say that A is i near in ν if νA0, ii ν-clan if A is a near grill, iii ν-closed if ν{{A}∪B}0, for all B⊆A⇒A ∈A, iv ν-cluster if A is a ν-closed ν-clan. → For any approach merotopic spaces X, νX and Y, νY , a map f : X Y is called a A ≤ A A∈P2 −1B ≥ B contraction if νY f νX , for all X , or equivalently νX f νY , for B∈P2 ≤ all Y . For any approach merotopies ν1 and ν2 on X, ν2 ν1 ν1 is finer than ν2, or ν2 is → coarser than ν1 if the identity mapping 1X : X, ν1 X, ν2 is a contraction see 19 . For standard definitions in the theory of categories we refer to 22, for approach spaces we refer to 14, and for lattices see 23. 3. Approach Merotopological Spaces In this section, we introduce approach merotopological spaces and establish some category- theoretic results for them. Lattice structure of the family of all approach merotopologies on X is also discussed. Definition 3.1. An approach merotopological space is a triple X, ν, cl, where ν is an approach merotopy on X and cl is a Kuratowski closure operator on X such that the following condition is satisfied: AM5 νclA ≥ νA, for all A∈P2X. We call ν to be an approach merotopology with respect to the closure operator cl on X. Example 3.2. i Let X, ν, cl be an approach merotopological space and let GX be the family P2 → ∞ of all grills on X. Define νg : X 0, as follows: A G A≺G G∈G ∀A ∈ P2 νg inf{ν : , X}, X. 3.1 Then νg is an approach merotopology with respect to cl on X see 19, 24 . ii Let X, cl be a topological space. Then X, νg , cl is an approach merotopological G → ∞ space, where νg is defined as in the above example in which ν : X 0, can be defined in the following ways for G∈GX. G G ∅ G {| | ∈G | | ℵ } a ν 0, if cl / ;andν sup cl G : G and cl G < 0 , otherwise. G G ∅ G | | b ν 0, if cl / ;andν supG∈G stack cl G , otherwise. 4 International Journal of Mathematics and Mathematical Sciences c νG0, if clG / ∅;andνG| clG|, otherwise. G G ∅ G | | d ν 0, if cl / ;andν infG∈sec clG G , otherwise. e νG0, if clB / ∅ for every finite subset B of G;andνGinf{|G| : G ∈ sec clG}, otherwise. f νG0, if clG / ∅ or each element of G is infinite; and νGsup{|G| : G ∈G | | ℵ } and G < 0 , otherwise. In this case, cl is a T1-closure operator on X. Observe that if X, cl is a symmetrical topological space i.e., x ∈ cl{y} ⇒ y ∈ { } ∈ cl x , for all x, y X , then clν cl in all of the above cases note that a T1-space is already a symmetrical topological space. → G → iii Let f : X, clX Y, clY be a closed and continuous map. Define ν : X ∞ G∈G G { ∈G} ∅ G 0, as follows: for X ,ν 0, if clY f G : G / ;andν | | infG∈sec clG G , otherwise. Then X, νg , clX is an approach merotopological space. Further ∈ ⇒ ∈ { } ∈ ∈ if y clY f x f x clY y for all y Y and x X, then clν clX. P2 → ∞ iv Let X, ν, cl be an approach merotopological space. Define νc : X 0, as follows: for A∈P2X, A B B⊆A B ℵ νc sup{ν : , | | < 0}.
Load more

Recommended publications
  • SOBER APPROACH SPACES 1. Introduction in This Article We Will
    SOBER APPROACH SPACES B. BANASCHEWSKI, R. LOWEN, C. VAN OLMEN 1. Introduction In this article we will introduce the notion of sobriety for approach spaces, modelled after the corresponding familiar concept in the case of topological spaces. Recall that the latter is closely linked to the relationship between the categories Top of topological spaces and Frm of frames, these being certain complete lattices which may be viewed as abstractly defined lat- tices of virtual open sets. More specifically, the connection between these categories is provided by a pair of contravariant functors, adjoint on the right, which assign to each topological space its frame of open subsets and to each frame its spectrum. Sober spaces are then those spaces for which the corresponding adjunction map is a homeomorphism. In the approach setting, the so-called regular function frame of an ap- proach space takes over the role of the frame of open sets in topology and accordingly we introduce an abstractly defined version of this, called an ap- proach frame. This leads to the contravariant functor R from the category AP of approach spaces to AFrm of approach frames and the spectrum func- tor Σ in the opposite direction. R and Σ are then adjoint on the right and an approach space X is called sober if the corresponding adjunction map X → ΣRX is an isomorphism. The purpose of this paper is to give an account of the basic facts con- cerning the functors R and Σ and to present a number of results about sobriety, including its reflectiveness and the relation between sobriety and completeness for uniform approach spaces.
    [Show full text]
  • A New Approach to the Study of Extended Metric Spaces 1
    Mathematica Aeterna, Vol. 3, 2013, no. 7, 565 - 577 A New Approach to the Study of Extended Metric Spaces Surabhi Tiwari Department of Mathematics Motilal Nehru National Institute of Technology Allahabad, U.P., India. Email: [email protected], [email protected] James F. Peters Computational Intelligence Laboratory Department of Electrical & Computer Engineering University of Manitoba Winnipeg, Manitoba R3T 51-V6, Canada. Email: [email protected] Abstract This paper studies the category theoretic aspects of ε-approach near- ness spaces and ε-approach merotopic spaces, respectively, ε ∈ (0, ∞], which are useful in measuring the degree of nearness (resemblance) of objects. Such structures measure the almost nearness of two collections of subsets of a nonempty set. The categories εANear and εAMer are shown to be full supercategories of various well-known categories, including the category sT op of symmetric topological spaces and con- tinuous maps, and the category Met∞ of extended metric spaces and nonexpansive maps. The results in this paper have important practical implications in the study of patterns in similar pictures. Mathematics Subject Classification: 54E05, 54E17, 54C25. Keywords: Approach merotopy, Nearness space, Topological space. 1 Introduction In [13], the two-argument ε-approach nearness was axiomatized for the pur- pose of presenting an alternative approach for completing extended metric spaces. This completion of extended metric spaces was motivated by the recent work on approach merotopic spaces in [7, 6, 4, 5, 16], (see also [2, 3]). In [15], 566 Surabhi Tiwari and James F. Peters ε-approach nearness was used to prove Niemytzki-Tychonoff theorem for sym- metric topological spaces.
    [Show full text]
  • The Notion of Closedness and D-Connectedness in Quantale
    THE NOTION OF CLOSEDNESS AND D-CONNECTEDNESS IN QUANTALE-VALUED APPROACH SPACES MUHAMMAD QASIM AND SAMED OZKAN¨ Abstract. In this paper, we characterize the local T0 and T1 separa- tion axioms for quantale-valued gauge space, show how these concepts are related to each other and apply them to L-approach space and L- approach system. Furthermore, we give the characterization of a closed point and D-connectedness in quantale-valued gauge space. Finally, we compare all these concepts with other. 1. Introduction Approach spaces have been introduced by Lowen [25, 26] to generalize metric and topological concepts, have many applications in almost all ar- eas of mathematics including probability theory [13], convergence theory [14], domain theory [15] and fixed point theory [16]. Due to its huge im- portance, several generalization of approach spaces appeared recently such as probabilistic approach spaces [18], quantale-valued gauge spaces [19], quantale-valued approach system [20] and quantale-valued approach w.r.t closure operators [24]. Recently, some quantale-valued approach spaces [21] are also characterized by using quantale-valued bounded interior spaces and bounded strong topological spaces which are commonly used by fuzzy math- ematicians. In 1991, Baran [2] introduced local separation axioms and notion of closed- ness in set-based topological categories which are used to define several dis- arXiv:1811.00767v1 [math.GN] 2 Nov 2018 tinct Hausdorff objects [4], T3 and T4 objects [6], regular, completely regular, normed objects [7], notion of compactness and minimality, perfectness [8] and closure operators in the sense of Guili and Dikranjan [17] in some well- known topological categories [9, 10].
    [Show full text]
  • The Missing Link in the Topology-Uniformity-Metric Triad
    Previous Up Next Citations From References: 143 From Reviews: 18 MR1472024 (98f:54002) 54-02 18B30 54Bxx 54Cxx 54Exx Lowen, R. [Lowen, Robert] (B-ANTW) FApproach spaces. The missing link in the topology-uniformity-metric triad. Oxford Mathematical Monographs. Oxford Science Publications. The Clarendon Press, Oxford University Press, New York, 1997. x+253 pp. $110.00. ISBN 0-19-850030-0 FEATURED REVIEW. This book is a landmark in the history of general topology. The author developed approach spaces in a series of papers from 1987 onwards and the book contains the results from those papers, presented in a systematic and definitive manner. Approach spaces form a supercategory of the categories of topological spaces and metric spaces, and the main purpose for their introduction was to fill some gaps concerning categorical aspects of metrizable spaces (see below for details). They not only solved these fundamental problems but also turned out to be interesting objects of study for their own sake. Moreover, they are useful in many areas such as probability theory, functional analysis, function spaces, hyperspaces, and probabilistic metric spaces. They also provide a proper setting for a unified treatment of certain topological and metric properties in the theory of approximation. Let us discuss a motivation. Suppose (X; d) is a metric space and δd(x; A) is the distance between x 2 X and A ⊂ X defined by inffd(x; a): a 2 Ag. We say that x is near A if δd(x; A) = 0. About ninety years back, F. Riesz showed that the concept of a continuous function can be defined by using this nearness relation between points and subsets.
    [Show full text]
  • Topological Games and Hyperspace Topologies
    Set-Valued Analysis 6: 187–207, 1998. 187 © 1998 Kluwer Academic Publishers. Printed in the Netherlands. Topological Games and Hyperspace Topologies LÁSZLÓ ZSILINSZKY Department of Mathematics and Computer Science, University of North Carolina, Pembroke, NC 28372, U.S.A. e-mail: [email protected] (Received: 29 September 1997; in final form: 25 May 1998) Abstract. The paper proposes a unified description of hypertopologies, i.e. topologies on the non- empty closed subsets of a topological space, based on the notion of approach spaces introduced by R. Lowen. As a special case of this description we obtain the abstract hit-and-miss, proximal hit-and-miss and a big portion of weak hypertopologies generated by gap and excess functionals (including the Wijsman topology and the finite Hausdorff topology), respectively. We also give char- acterizations of separation axioms T0;T1;T2;T3 and complete regularity as well as of metrizability of hypertopologies in this general setting requiring no additional conditions. All this is done to provide a background for proving the main results in Section 4, where we apply topological games (the Banach–Mazur and the strong Choquet game, respectively) to establish various properties of hypertopologies; in particular we characterize Polishness of hypertopologies in this general setting. Mathematics Subject Classifications (1991): 54B20; 54A05, 54D10, 54D15, 54E35, 54E52, 90D44. Key words: approach spaces, (proximal) hit-and-miss topologies, weak hypertopologies, separation axioms, metrizability, Banach–Mazur game, strong Choquet game, (strongly) α-favorable spaces, Polish spaces. 0. Introduction There has been an impressive growth in the number of recently introduced topolo- gies on the hyperspace (i.e.
    [Show full text]
  • Approach Merotopological Spaces and Their Completion
    Hindawi Publishing Corporation International Journal of Mathematics and Mathematical Sciences Volume 2010, Article ID 409804, 16 pages doi:10.1155/2010/409804 Research Article Approach Merotopological Spaces and their Completion Mona Khare and Surabhi Tiwari Department of Mathematics, University of Allahabad, Allahabad 211002, India Correspondence should be addressed to Mona Khare, [email protected] Received 24 July 2009; Accepted 13 April 2010 Academic Editor: Richard Wilson Copyright q 2010 M. Khare and S. Tiwari. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. This paper introduces the concept of an approach merotopological space and studies its category- theoretic properties. Various topological categories are shown to be embedded into the category whose objects are approach merotopological spaces. The order structure of the family of all approach merotopologies on a nonempty set is discussed. Employing the theory of bunches, bunch completion of an approach merotopological space is constructed. The present study is a unified look at the completion of metric spaces, approach spaces, nearness spaces, merotopological spaces, and approach merotopological spaces. 1. Introduction Some of the applications of nearness-like structures within topology are unification, extensions, homology, and connectedness. The categories of R0-topological spaces, uniform spaces 1, 2, proximity spaces 2, 3, and contiguity spaces 4, 5 are embedded into the category of nearness spaces. The study of proximity, contiguity, and merotopic spaces in the more generalized setting of L-fuzzy theory can be seen in 6–13.In14, the notion of an approach space was introduced via different equivalent set of axioms to measure the degree of nearness between a set and a point.
    [Show full text]
  • Quantale-Valued Generalizations of Approach Spaces: L-Approach Systems
    http://topology.auburn.edu/tp/ Volume 51, 2018 Pages 253{276 http://topology.nipissingu.ca/tp/ Quantale-valued generalizations of approach spaces: L-approach systems by Gunther Jager¨ Electronically published on January 18, 2018 This file contains only the first page of the paper. The full version of the paper is available to Topology Proceedings subscribers. See http://topology.auburn.edu/tp/subscriptioninfo.html for information. Topology Proceedings Web: http://topology.auburn.edu/tp/ Mail: Topology Proceedings Department of Mathematics & Statistics Auburn University, Alabama 36849, USA E-mail: [email protected] ISSN: (Online) 2331-1290, (Print) 0146-4124 COPYRIGHT ⃝c by Topology Proceedings. All rights reserved. http://topology.auburn.edu/tp/ http://topology.nipissingu.ca/tp/ TOPOLOGY PROCEEDINGS Volume 51 (2018) Pages 253-276 E-Published on January 18, 2018 QUANTALE-VALUED GENERALIZATIONS OF APPROACH SPACES: L-APPROACH SYSTEMS GUNTHER JÄGER Abstract. We define and study quantale-valued approach sys- tems. We show that the resulting category is topological and study its relation to other categories of quantale-valued generalizations of approach spaces, such as the categories of quantale-valued approach spaces and of quantale-valued gauge spaces. We pay particular at- tention to the probabilistic case. 1. Introduction The category of approach spaces, introduced in [13], is a common su- percategory of the categories of metric and topological spaces. The theory of these spaces is far developed and has many applications as is demon- strated in e.g. [14, 15]. In simple terms one may say, that the theory of approach spaces is “metrical” in the sense that an approach space is of- ten either defined by a point-set distance function or a suitable family of metrics (a so-called gauge) or by families of “local distances” (so-called ap- proach systems).
    [Show full text]
  • Approach Theory Meets Probability Theory ✩ ∗ B
    View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector Topology and its Applications 158 (2011) 836–852 Contents lists available at ScienceDirect Topology and its Applications www.elsevier.com/locate/topol Approach theory meets probability theory ✩ ∗ B. Berckmoes , R. Lowen, J. Van Casteren Department of Mathematics and Computer Science, University of Antwerp, Belgium article info abstract MSC: In this paper we reconsider the basic topological and metric structures on spaces of 60A05 probability measures and random variables, such as e.g. the weak topology and the total 60G05 variation metric, replacing them with more intrinsic and richer approach structures. We 60G07 comprehensibly investigate the relationships among, and basic facts about these structures, 62P99 and prove that fundamental results, such as e.g. the portmanteau theorem and Prokhorov’s 65D99 theorem, can be recaptured in a considerably stronger form in the new setting. © Keywords: 2011 Elsevier B.V. All rights reserved. Approach space Distance Weak topology Polish space Probability measure Law Random variable Weak compactness Tightness Prokhorov metric Ky-Fan metric Atsuji space 1. Introduction In this paper we reconsider the basic topological and metric structures on spaces of probability measures and spaces of random variables. It turns out that many topological structures and metrics used in the literature [4,6,16,19,23] are merely respectively the topological coreflections and the metric coreflections of more natural and structurally richer approach struc- tures. For the sake of simplicity, throughout the paper we will consider a Polish space S and the set P(S) of probability measures on it.
    [Show full text]
  • Mappings Between Distance Sets Or Spaces
    Mappings between distance sets or spaces Vom Fachbereich Mathematik der Universität Hannover zur Erlangung des Grades Doktor der Naturwissenschaften Dr. rer. nat. genehmigte Dissertation von Dipl.-Math. Jobst Heitzig geboren am 17. Juli 1972 in Hannover 2003 diss.tex; 17/02/2003; 8:34; p.1 Referent: Prof. Dr. Marcel Erné Korreferent: Prof. Dr. Hans-Peter Künzi Tag der Promotion: 16. Mai 2002 diss.tex; 17/02/2003; 8:34; p.2 Zusammenfassung Die vorliegende Arbeit hat drei Ziele: Distanzfunktionen als wichtiges Werkzeug der allgemeinen Topologie wiedereinzuführen; den Gebrauch von Distanzfunktionen auf den verschiedensten mathematischen Objekten und eine Denkweise in Begriffen der Abstandstheorie anzuregen; und schliesslich spezifischere Beiträge zu leisten durch die Charakterisierung wichtiger Klassen von Abbildungen und die Verallgemeinerung einiger topologischer Sätze. Zunächst werden die Konzepte des ,Formelerhalts‘ und der ,Übersetzung von Abständen‘ benutzt, um interessante ,nicht-topologische‘ Klassen von Abbildungen zu finden, was zur Charakterisierung vieler bekannter Arten von Abbildungen mithilfe von Abstandsfunktionen führt. Nachdem dann eine ,kanonische‘ Methode zur Konstruktion von Distanzfunktionen angegeben wird, entwickele ich einen geeigneten Begriff von ,Distanzräumen‘, der allgemein genug ist, um die meisten topologischen Strukturen induzieren zu können. Sodann werden gewisse Zusammenhänge zwischen einigen Arten von Abbildungen bewiesen, wie z. B. dem neuen Konzept ,streng gleichmässiger Stetigkeit‘. Es folgt eine neuartige
    [Show full text]
  • TOPOLOGICAL PROPERTIES of NON-ARCHIMEDEAN APPROACH SPACES 1. Introduction
    Theory and Applications of Categories, Vol. 32, No. 41, 2017, pp. 1454{1484. TOPOLOGICAL PROPERTIES OF NON-ARCHIMEDEAN APPROACH SPACES EVA COLEBUNDERS AND KAREN VAN OPDENBOSCH Abstract. In this paper we give an isomorphic description of the category of non- Archimedian approach spaces as a category of lax algebras for the ultrafilter monad and an appropriate quantale. Non-Archimedean approach spaces are characterised as those approach spaces having a tower consisting of topologies. We study topological properties p; for p compactness and Hausdorff separation along with low-separation properties, regularity, normality and extremal disconnectedness and link these properties to the condition that all or some of the level topologies in the tower have p. A compactification technique is developed based on Shanin's method. 1. Introduction In monoidal topology [Hofmann, Seal, Tholen (eds.), 2014] starting from the quantale op P+ = [0; 1] ; +; 0 where the natural order ≤ of the extended real halfline [0; 1] is reversed so that 0 = > and 1 = ?, based on Lawvere's description of quasi-metric spaces as categories enriched over the extended real line [Lawvere 1973], one obtains an isomorphism between the category qMet of extended quasi-metric spaces with non- expansive maps and P+ -Cat, the category of P+ -spaces (X; a), where X is a set and a : X−!7 X is a transitive and reflexive P+ -relation, meaning it is a map a : X × X ! P+ satisfying a(x; y) ≤ a(x; z) + a(z; y) and a(x; x) = 0 for all x; y; z in X; with morphisms 0 0 0 f :(X; a) ! (Y; b) of P+ -Cat, maps satisfying b(f(x); f(x )) ≤ a(x; x ) for all x; x in X: Our interest goes to the larger category App of approach spaces and contractions, which contains qMet as coreflectively embedded full subcategory and which is better behaved than qMet with respect to products of underlying topologies.
    [Show full text]
  • Metric, Topology, Uniformity, Approach Structure
    Applied Categorical Structures 12: 127–154, 2004. 127 © 2004 Kluwer Academic Publishers. Printed in the Netherlands. One Setting for All: Metric, Topology, Uniformity, Approach Structure MARIA MANUEL CLEMENTINO1, DIRK HOFMANN2 and WALTER THOLEN3 1Departamento de Matemática, Universidade de Coimbra, 3001-454 Coimbra, Portugal. e-mail: [email protected] 2Departamento de Matemática, Universidade de Aveiro, 3810-193 Aveiro, Portugal. e-mail: [email protected] 3Department of Mathematics and Statistics, York University, Toronto, Ontario, Canada M3J 1P3. e-mail: [email protected] (Received: 21 November 2002; accepted: 4 December 2002) Abstract. For a complete lattice V which, as a category, is monoidal closed, and for a suitable Set-monad T we consider (T, V)-algebras and introduce (T, V)-proalgebras, in generalization of Lawvere’s presentation of metric spaces and Barr’s presentation of topological spaces. In this lax- algebraic setting, uniform spaces appear as proalgebras. Since the corresponding categories be- have functorially both in T and in V, one establishes a network of functors at the general level which describe the basic connections between the structures mentioned by the title. Categories of (T, V)-algebras and of (T, V)-proalgebras turn out to be topological over Set. Mathematics Subject Classifications (2000): 18C20, 18B30, 54E15. Key words: V-matrix, V-promatrix, (T, V)-algebra, (T, V)-proalgebra, co-Kleisli composition, or- dered set, metric space, topological space, uniform space, approach space, prometric space, protopo- logical
    [Show full text]
  • Scott Approach Distance on Metric Spaces
    Scott approach distance on metric spaces∗ Dedicated to Robert Lowen on his 70th birthday Wei Li,† Dexue Zhang‡ Abstract The notion of Scott distance between points and subsets in a metric space, a metric analogy of the Scott topology on an ordered set, is introduced, making a metric space into an approach space. Basic properties of Scott distance are investigated, including its topological coreflection and its relation to injective T0 approach spaces. It is proved that the topological coreflection of the Scott distance is sandwiched between the d-Scott topology and the generalized Scott topology; and that every injective T0 approach space is a cocomplete and continuous metric space equipped with its Scott distance. Keywords Metric space, Approach space, Scott distance, d-Scott topology, Gen- eralized Scott topology, Algebraic metric space, Continuous metric space, Injective approach space MSC(2010) 06B35, 18B30, 18B35, 54A05, 54E35 1 Introduction In 1989, Lowen [24] introduced approach spaces as a common extension of metric spaces and topological spaces. As explored in the monograph [26], approach spaces are closely related to many disciplines in mathematics, e.g. topology, analysis, probability, domain theory and etc. This paper focuses on one aspect of approach spaces, that is, their relation to metric spaces from the viewpoint of domain theory. On one hand, following Lawvere [22], metric spaces (not necessarily symmetric) can be thought of as ordered sets valued in the closed category ([0, ∞]op, +). This point of view has led to the theory of quantitative domains, initiated by Smyth [29, 30], with metric spaces as core objects, see e.g.
    [Show full text]