DOCSLIB.ORG

Mappings Between Distance Sets Or Spaces

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read 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 Charakterisierung der Ähnlichkeitsabbildungen zwis- chen Euklidischen Räumen. Die Dissertation schliesst mit einigen Verallgemeinerungen bekannter Vervollständigungskonstruktionen und wichtiger Fixpunktsätze, und einer kurzen Studie über Techniken der Visualisierung von Abständen. Abstract The aim of this thesis is threefold: to reinstate distance functions as a principal tool of general topology; to promote the use of distance functions on various mathematical objects and a thinking in terms of distances also in non-topological contexts; and to make more specific contributions by characterizing important classes of mappings and generalizing some important topological results. I start by using the key concepts of ‘preservation of formulae' and ‘translation of distances' to extract interesting ‘non-topological’ classes of mappings, which leads to the characterization of many well-known types of mappings in terms of distance functions. After giving a ‘canonical’ method for constructing distance functions, a suitable notion of ‘distance spaces' will be developed, general enough to induce most topological structures. Then certain relationships between many kinds of mappings are proved, including the new concept of ‘strong uniform continuity', followed by a new characterization of the similarity maps between Euclidean spaces. The thesis closes with some generalizations of completions and fixed point theorems, and a short, self-contained study of distance visualization techniques. Schlagworte: Distanzfunktion, Abbildung, Topologie Key words: distance function, mapping, topology diss.tex; 17/02/2003; 8:34; p.3 Danksagung Dank geht an Herrn Dipl.-Math. Lars Ritter und Dr. Daniel Frohn für Diskussionen und gelegentliche Korrekturlesearbeiten, und an Dr. Jürgen Reinhold für die erfolgreiche gemeinsame Zählerei. Dr. Matthias Kriesell danke ich sehr für die Bereitschaft, sich immer wieder auch unausgegorene Ideen anzuhören und meinen Gehirnknoten weitere Kanten hinzuzufügen. Vor allen anderen gilt mein Dank meinem Doktorvater Prof. Dr. Marcel Tonio Erné, mit dem ich unzählbare Stunden fruchtbaren Gedankenaustauschs teilen durfte und dem es gelang, unermüdlich meinen mathematischen Horizont zu erweitern. Ohne seine Unterstützung wäre eine solche breitangelegte Studie wohl kaum möglich gewesen. diss.tex; 17/02/2003; 8:34; p.4 Contents A GENERAL CONCEPT OF DISTANCE 1. Distance sets 6 Definitions 6 Real distances 7 Multi-real distances 12 −− −− −− Distances in classical algebraic structures 15 Some other concepts of ‘generalized metric' 23 −− −− 2. Mappings 25 Distance sets with the same value monoid 25 Translating distances of different type 29 −− −− Comparing distance functions on the same set 40 −− BACK TO THE ROOTS OF TOPOLOGY 3. Convergence and closure 48 Converging to the right class of structures 48 Open and closed; filters and nets 54 −− −− Distances in point-free situations, and hyperspaces 62 −− 4. More on mappings 68 Topological properties of maps 68 Maps with both topological and non-topological properties 72 −− −− Finest distance structures 82 −− 5. Fundamental nets and completeness 96 Fundamental nets and Cauchy-filters 96 Notions of completeness 99 Distances between nets 102 −− −− −− Some completions 105 −− 6. Fixed points 110 Banach: Fixed points of Lipschitz-continuous maps 110 −− Sine–Soardi: Fixed points of contractive maps 114 Brouwer: Fixed points of continuous maps 115 −− −− APPENDIX Visualization of distances 122 Additional proofs 137 References 140 Indices 144 diss.tex; 17/02/2003; 8:34; p.5 diss.tex; 17/02/2003; 8:34; p.6 INTRODUCTION −−− Railroad, telephone, bicycle, automobile, air plane, and cinema revolutionized the sense of distance. [::: ] Distances depended on the effect of memory, the force of emotions, and the passage of time. Stephen Kern, The Culture Of Time And Space 1880–1918 In everyday language, ‘distance’ has always been something more general than the length of a segment in some geometrical space. Instead, the concept of ‘near’ and ‘far’ is one of the more important categories in human thinking. Extracting the abstract idea from the physical phenomenon, we speak of the growing distance to an old friend, of how near we are to reach a certain goal, or how far from being jealous. It is important that quite often the interesting question is not ‘‘how much is in between x and y'' but rather ‘‘how much is needed to get from x to y''. This somewhat dynamical interpretation of distance differs from the geometrical one in that it does not imply any symmetry, positivity, or strictness a priori. It is only natural when mathematicians, too, think of their objects as being related by the one or other kind of distance and how surprising is it that we still −− require mathematical distances to be real-valued, mostly symmetric, and non- negative? Before 1900, mathematical distances had beed used mainly in geometry and as a measure of difference between real numbers or functions. They had also played an important role for the clarification of the notion of ‘real number' itself, which in turn was a strong impetus for the development of topology. In the beginning of the last century, when Fréchet [Fré05, Fré06, Fré28] and Hausdorff [Hau14, Hau27, Hau49] initiated the axiomatic study of distances in the general setting of metric instead of geometric spaces, the real numbers were 1 2 therefore the natural candidates for the values of a distance function. Complex numbers or real vectors, being imaginable alternatives, would most certainly not have been considered suitable because of the difficulties in ordering such entities given that partial orders had not received much attention at that time. −− On the other hand, rational numbers had already long been known to be too special because of their lacking completeness. Just as in case of measure theory, it is therefore not surprising that the theoretical treatment of distances was dominated by a paradigm of using real numbers. Although, from the beginning, general topology was far more than the study of metric spaces, the question of which topological spaces can be endowed with a suitable metric, known as the ‘metrization problem', remained important. This was not only because metric spaces had very nice topological properties, mostly inherited from even nicer properties of the real numbers themselves, but also since the idea of distance remained a principal intuition in building new topological concepts, and because topological spaces alone had not enough structure to formulate certain interesting notions. For example, Lipschitz- and uniform continuity, or completeness, being of great importance in real analysis, cannot be expressed in terms of open sets alone. This motivated the search for suitable structural additives to general topo- logical spaces, which could well have led to an early study of substantially more general distance functions than real metrics. But despite only a few attempts in the latter direction, the researchers in this field soon focused on systems of subsets instead, ending up with the notion of ‘uniform space' (cf. [BHH98]). However, there were situations when distances had a great chance of being reconsidered passing virtually unnoticed. Van Dantzig [vD32], for −− instance, defined fundamental sequences in a topological group, using Menger's ‘Gruppenmetrik’ [Men31] without recognizing it as a distance function. Even more surprisingly, Kelley essentially proved that every uniformity (even every quasi-uniformity) comes from a family of real-valued distance functions [Kel55], but despite the popularity of his classical textbook, the theory of uniform spaces did not yet enter a possibly fruitful engagement with a theory of vector-valued metrics. The aim of this thesis is threefold: to reinstate distance functions as a principal tool of general topology; to promote the use of distance functions on various mathematical objects and a thinking in terms of distances also in non-topological contexts; and to make
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]
  • 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]
  • Elementary Properties of Ordered Abelian Groups
    ELEMENTARY PROPERTIES OF ORDERED ABELIAN GROUPS BY ABRAHAM ROBINSON and ELIAS ZAKON Introduction. A complete classification of abelian groups by their ele- mentary properties (i.e. properties that can be formalized in the lower predi- cate calculus) was given by Szmielew [9]. No such attempt, however, has so far been made with respect to ordered groups. In the present paper the classification by elementary properties will be carried out for all archimedean ordered abelian groups and, moreover, for a certain more general class of groups which we shall call regularly ordered. Simultaneously, necessary and sufficient conditions will be established for two such groups to be elementarily equivalent, i.e. to have all their elementary properties in common (see Theo- rem 4.7). By a complete classification of a class of groups we mean its partition into disjoint subclasses in such a way that two groups belong to one subclass if, and only if, they are elementarily equivalent. This goal will be attained by setting up a series of (finite or infinite) complete systems of axioms, each sys- tem defining a certain subclass of ordered abelian groups. The notation and terminology of [8] will be used throughout. In par- ticular, the concept of model-completeness introduced in [8] and [7] will be applied. A novelty feature of the present paper is that the method based on model-completeness will be combined with what we shall call "adjunction of new relations." To illustrate the usefulness of the method, we shall also apply it to give new simplified proofs of some theorems by Langford and Tarski on the completeness of certain systems of axioms referring to ordered sets.
    [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 Computable Dimension of Ordered Abelian Groups
    CORE Metadata, citation and similar papers at core.ac.uk Provided by Elsevier - Publisher Connector Advances in Mathematics 175 (2003) 102–143 http://www.elsevier.com/locate/aim The computable dimension of ordered abelian groups Sergey S. Goncharov,a Steffen Lempp,b and Reed Solomonc,* a Institute of Mathematics, Siberian Branch, Russian Academy of Sciences, Russia b Department of Mathematics, University of Wisconsin-Madison, Madison, WI 53706, USA c Department of Mathematics, University of Connecticut, Storrs, CT 06269, USA Received 27 February 2001; accepted 14 February 2002 Communicated by H. Jerome Keisler Abstract Let G be a computable ordered abelian group. We show that the computable dimension of G is either 1 or o; that G is computably categorical if and only if it has finite rank, and that if G has only finitely many Archimedean classes, then G has a computable presentation which admits a computable basis. r 2003 Elsevier Science (USA). All rights reserved. MSC: 03D; 06F Keywords: Computability; Computable model theory; Ordered groups 1. Introduction In this article, we examine countable ordered abelian groups from the perspective of computable algebra. We begin with the definition and some examples of ordered abelian groups. Definition 1.1. An ordered abelian group is a pair ðG; pGÞ; where G is an abelian group and pG is a linear order on G such that if a pG b; then a þ g pG b þ g for all gAG: *Correspondingauthor. E-mail addresses: [email protected] (S.S. Goncharov), [email protected] (S. Lempp), [email protected] (R.
    [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]
  • Summary on Non-Archimedean Valued Fields 3
    Contemporary Mathematics Volume 704, 2018 http://dx.doi.org/10.1090/conm/704/14158 Summary on non-Archimedean valued fields Angel Barr´ıa Comicheo and Khodr Shamseddine Abstract. This article summarizes the main properties of ultrametric spaces, valued fields, ordered fields and fields with valuations of higher rank, highlight- ing their similarities and differences. The most used non-Archimedean valued fields are reviewed, like a completion in the case of the p-adic numbers fields and the Levi-Civita fields, or like an algebraic closure as is sometimes the case of the Puiseux series fields. Also a study of spherically complete valued fields is presented and their characterization as maximally complete fields is given where the Hahn fields and the Mal’cev-Neumann fields (their p-adic ana- logues) play an important role as “spherical completions”. Finally several of the metric, topological, algebraic and order properties of the most used non- Archimedean valued fields are collected in a catalog where we can appreciate the analogy between fields that have the same characteristic as their residue class fields and fields that do not satisfy this property. Introduction. When a graduate student or any mathematician in general begin their jour- ney in non-Archimedean Analysis, the freedom of choice for the base valued field presents different cases to consider, most of which do not occur when we work with the fields of real or complex numbers. The existence of discrete valuations, or the possible non-existence of vectors of norm 1 sometimes lead to the use of new procedures for proofs of results in Functional Analysis.
    [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]
  • Or Natural) Density D(A)Forsetsa of Natural Numbers Is a Central Tool in Number Theory: |A ∩ [1,N]| D(A) = Lim (Provided the Limit Exists
    PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 138, Number 8, August 2010, Pages 2657–2665 S 0002-9939(10)10351-7 Article electronically published on April 1, 2010 FINE ASYMPTOTIC DENSITIES FOR SETS OF NATURAL NUMBERS MAURO DI NASSO (Communicated by Julia Knight) Abstract. By allowing values in non-Archimedean extensions of the unit in- terval, we consider finitely additive measures that generalize the asymptotic density. The existence of a natural class of such “fine densities” is independent of ZFC. Introduction The asymptotic (or natural) density d(A)forsetsA of natural numbers is a central tool in number theory: |A ∩ [1,n]| d(A) = lim (provided the limit exists). n→∞ n In many applications, it is useful to consider suitable extensions of d that are defined for all subsets. Among the most relevant examples, there are the upper and lower density,theSchnirelmann density,andtheupper Banach density (see e.g. [9], [7], [13]). Several authors investigated the general problem of densities, i.e. the possibility of constructing finitely additive measures that extend asymptotic density to all subsets of the natural numbers and that satisfy some additional properties (see e.g. [2], [11], [12] and [1]). Recently, generalized probabilities have been introduced that take values into non-Archimedean rings (see e.g. [8] and [10]). In this paper we pursue the idea of refining the notion of density by allowing values into a non-Archimedean extension of the unit interval. To this aim, we introduce a notion of “fine density” as a suitable finitely additive function on P(N) that gives a non-zero (infinitesimal) measure even to singletons.
    [Show full text]