DOCSLIB.ORG

Continuous Closure, Natural Closure, and Axes Closure

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Continuous Closure, Natural Closure, and Axes Closure CONTINUOUS CLOSURE, AXES CLOSURE, AND NATURAL CLOSURE NEIL EPSTEIN AND MELVIN HOCHSTER Abstract. Let R be a reduced affine C-algebra, with corresponding affine algebraic set X.Let (X)betheringofcontinuous(Euclideantopology)C- C valued functions on X.Brennerdefinedthecontinuous closure Icont of an ideal I as I (X) R.Healsointroducedanalgebraicnotionofaxes closure Iax that C \ always contains Icont,andaskedwhethertheycoincide.Weextendthenotion of axes closure to general Noetherian rings, defining f Iax if its image is in 2 IS for every homomorphism R S,whereS is a one-dimensional complete ! seminormal local ring. We also introduce the natural closure I\ of I.Oneof many characterizations is I\ = I+ f R : n>0withf n In+1 .Weshow { 2 9 2 } that I\ Iax,andthatwhencontinuousclosureisdefined,I\ Icont Iax. ✓ ✓ ✓ Under mild hypotheses on the ring, we show that I\ = Iax when I is primary to a maximal ideal, and that if I has no embedded primes, then I = I\ if and only if I = Iax,sothatIcont agrees as well. We deduce that in the @f polynomial ring [x1,...,xn], if f =0atallpointswhereallofthe are C @xi 0, then f ( @f ,..., @f )R.WecharacterizeIcont for monomial ideals in 2 @x1 @xn polynomial rings over C,butweshowthattheinequalitiesI\ Icont and ✓ Icont Iax can be strict for monomial ideals even in dimension 3. Thus, Icont ax✓ and I need not agree, although we prove they are equal in C[x1,x2]. Contents 1. Introduction 2 2. Properties of continuous closure 4 3. Seminormal rings 8 4. Axes closure and one-dimensional seminormal rings 13 5. Special and inner integral closure, and natural closure 18 6. The ideal generated by the partial derivatives 25 7. Naturally closed primary ideals are axes closed 26 8. Multiplying by invertible ideals and rings of dimension 2 33 9. A negative example and a fiber criterion for exclusion from the continuous closure 35 10. Mixed natural closure and continuous closure for monomial ideals in polynomial rings 37 11. A bigger axes closure 44 Acknowledgments 47 Date:July27,2016. 2010 Mathematics Subject Classification. Primary 13B22, 13F45; Secondary 13A18, 46E25, 13B40, 13A15. Key words and phrases. continuous closure, axes closure, natural closure, seminormal ring. The second named author is grateful for support from the National Science Foundation, grants DMS-0901145 and DMS-1401384. 1 2 NEILEPSTEINANDMELVINHOCHSTER References 47 1. Introduction Holger Brenner [Bre06] introduced a new closure operation on ideals in finitely generated C-algebras called continuous closure, and asks whether it is the same as an algebraic notion called axes closure that he introduces. He proves this for ideals in a polynomial ring that are primary to a maximal ideal and generated by monomials. We shall relate this closure to some variant notions of integral closure, special part of the integral closure over a local ring, introduced in [Eps10], and inner integral closure, a notion explored here that exists without an explicit name in the literature, and also to a notion we introduce called natural closure. We shall prove that if I is an unmixed ideal in any affine C-algebra, then I is continuously closed if and only if it is axes closed. See Theorem 7.8, Corollary 7.14, and Corollary 7.15. We also provide further conditions under which continuous closure equals axes closure or natural closure. In consequence we can prove, for example, that if f is a polynomial in C[x1,...,xn] that vanishes wherever its partial derivatives all vanish, then there are continuous n functions gj from C C such that ! n @f f = g . j @x j=1 j X See Theorem 6.1. On the other hand, we show that continuous closure is sometimes strictly smaller than axes closure. Indeed, in 9, we give an example (followed by a method of § generating such examples) of a monomial ideal in a polynomial ring over C which is continuously closed but not axes closed. After hearing the second named author give a talk on the results of this paper, Koll´ar[Kol12] studied continuous closure in the context of coherent sheaves on schemes over C and has given an algebraic characterization that permits the notion of continuous closure to be defined in a larger context. In a further paper [FK13], continuous closure is studied over topological fields other than C, particularly for the field of real numbers. Let R be a finitely generated C-algebra. Map a polynomial ring C[x1,...,xn] ⇣ R onto R as C-algebras. Let A C[x1,...,xn] be the kernel ideal, and let X ✓ be the set of points in Cn where all elements of A vanish. X may be identified with the set of maximal ideals of R.ThenX has a Euclidean topology, and the topological space X is independent of the presentation of R.Welet (Y ) denote the ring of complex-valued continuous functions on any space Y . PolynomialC functions n on C ,whenrestrictedtoX,yieldaringC[X] which is isomorphic to Rred,the original ring modulo the ideal N of nilpotents. (Nothing will be lost in the sequel if we restrict attention to reduced rings R (i.e., rings without nonzero nilpotents).) Thus, we have a C-homomorphism R (X)whichisinjectivewhenR is reduced. The continuous closure of I R, denoted! C Icont, is the contraction of I (X)toR. That is, if I =(f ,...,f )R✓,thenf Icont precisely when there areC continuous 1 m 2 functions gi : X C such that ! f = g f + + g f , |X 1 1|X ··· m m|X CONTINUOUS CLOSURE, AXES CLOSURE, AND NATURAL CLOSURE 3 where h indicates the image of h R in (X). Henceforth, we focus on the case |X 2 C where R is reduced, and omit X from the notation. However, we can state many of the results without this hypothesis:| one can typically pass at once in the proofs to the case where the ring is reduced. In this paper we study this closure and several other closures that are related, obtaining satisfying answers to many questions that were open even for polynomial rings. Let L be an algebraically closed field. We are especially concerned with the case where L = C is the complex numbers. A finitely generated L-algebra R is called a ring of axes over L if it is one-dimensional, reduced, and either smooth, with just one irreducible component, or else is such that the corresponding algebraic set is the union of n smooth irreducible curves, and there is a unique singular point, which is the intersection of any two of the components, such that the completion of the local ring at that point is isomorphic with L[[ x1,...,xn]] /(xixj 1 i<j n). We now restrict to the case of the complex numbers. In [Bre06]| Brenner obtains a structure theorem for the ideals of such a completed local ring that enables him to prove that in a ring of axes over C, for every ideal I, I = Icont.Theaxes closure Iax of an ideal I of R is defined to be the set of elements r such that for every C- homomorphism R S,whereS is a ring of axes, one has r IS. (Here as in the sequel, we use the! standard abuse of notation r IS to mean2 that '(r) '(I)S, where ' : R S is the map from R to S.) The2 results of [Bre06] imply2 that Icont Iax in! general, and that they agree for ideals of polynomial rings that are primary✓ to maximal ideals and are generated by monomials. As mentioned above, we prove here that continuous closure coincides with axes closure for all ideals of affine C-algebras that are primary to a maximal ideal, and in many other cases. We also show that an unmixed ideal (one that has no embedded primes) is axes closed if and only if it is continuously closed, and that there exist continuously closed ideals which are not axes closed, which answers a question raised by Brenner. In 3 we prove that an element r of an affine C-algebra is in the axes closure of I § R if and only if x IS for every homomorphism of R to an excellent (respectively,✓ complete) Noetherian2 one-dimensional seminormal ring S.Weuse the latter definition to extend the notion of axes closure to all Noetherian rings. See Corollaries 4.2 and 4.4 and Definition 4.3. Here is a brief sketch of the contents of the paper: In 2, we discuss some important properties of continuous closure that we will need.§ Some of this material is reviewed from [Bre06], but in some cases we need sharper or more general results. 3 is devoted to seminormal rings and their connec- tions to continuous and axes closures.§ In 4, we extend the definition of axes closure to general Noetherian rings, characterizing§ it by maps to excellent one-dimensional seminormal rings, and we show that this agrees with the original definition in Bren- ner’s setting. In 5 we discuss the concepts of special and inner integral closure, and introduce the notion§ of natural closure. We also introduce the notion of I-relevant ideals, which are used to characterize when an ideal is naturally closed, and which play a key role in proving the results of 7. We show that the natural closure is contained in the axes closure and, wherever§ it is defined, the continuous closure. This “traps” continuous closure between two algebraically defined closures. This is the main tool used in 7 to prove results on when axes closure and continuous closure agree. § 4 NEILEPSTEINANDMELVINHOCHSTER One of the main results of 6 has already been stated in the second paragraph of this Introduction.
Load more

Recommended publications
  • Basic Properties of Filter Convergence Spaces
    Basic Properties of Filter Convergence Spaces Barbel¨ M. R. Stadlery, Peter F. Stadlery;z;∗ yInstitut fur¨ Theoretische Chemie, Universit¨at Wien, W¨ahringerstraße 17, A-1090 Wien, Austria zThe Santa Fe Institute, 1399 Hyde Park Road, Santa Fe, NM 87501, USA ∗Address for corresponce Abstract. This technical report summarized facts from the basic theory of filter convergence spaces and gives detailed proofs for them. Many of the results collected here are well known for various types of spaces. We have made no attempt to find the original proofs. 1. Introduction Mathematical notions such as convergence, continuity, and separation are, at textbook level, usually associated with topological spaces. It is possible, however, to introduce them in a much more abstract way, based on axioms for convergence instead of neighborhood. This approach was explored in seminal work by Choquet [4], Hausdorff [12], Katˇetov [14], Kent [16], and others. Here we give a brief introduction to this line of reasoning. While the material is well known to specialists it does not seem to be easily accessible to non-topologists. In some cases we include proofs of elementary facts for two reasons: (i) The most basic facts are quoted without proofs in research papers, and (ii) the proofs may serve as examples to see the rather abstract formalism at work. 2. Sets and Filters Let X be a set, P(X) its power set, and H ⊆ P(X). The we define H∗ = fA ⊆ Xj(X n A) 2= Hg (1) H# = fA ⊆ Xj8Q 2 H : A \ Q =6 ;g The set systems H∗ and H# are called the conjugate and the grill of H, respectively.
    [Show full text]
  • 7.2 Binary Operators Closure
    last edited April 19, 2016 7.2 Binary Operators A precise discussion of symmetry benefits from the development of what math- ematicians call a group, which is a special kind of set we have not yet explicitly considered. However, before we define a group and explore its properties, we reconsider several familiar sets and some of their most basic features. Over the last several sections, we have considered many di↵erent kinds of sets. We have considered sets of integers (natural numbers, even numbers, odd numbers), sets of rational numbers, sets of vertices, edges, colors, polyhedra and many others. In many of these examples – though certainly not in all of them – we are familiar with rules that tell us how to combine two elements to form another element. For example, if we are dealing with the natural numbers, we might considered the rules of addition, or the rules of multiplication, both of which tell us how to take two elements of N and combine them to give us a (possibly distinct) third element. This motivates the following definition. Definition 26. Given a set S,abinary operator ? is a rule that takes two elements a, b S and manipulates them to give us a third, not necessarily distinct, element2 a?b. Although the term binary operator might be new to us, we are already familiar with many examples. As hinted to earlier, the rule for adding two numbers to give us a third number is a binary operator on the set of integers, or on the set of rational numbers, or on the set of real numbers.
    [Show full text]
  • 3. Closed Sets, Closures, and Density
    3. Closed sets, closures, and density 1 Motivation Up to this point, all we have done is define what topologies are, define a way of comparing two topologies, define a method for more easily specifying a topology (as a collection of sets generated by a basis), and investigated some simple properties of bases. At this point, we will start introducing some more interesting definitions and phenomena one might encounter in a topological space, starting with the notions of closed sets and closures. Thinking back to some of the motivational concepts from the first lecture, this section will start us on the road to exploring what it means for two sets to be \close" to one another, or what it means for a point to be \close" to a set. We will draw heavily on our intuition about n convergent sequences in R when discussing the basic definitions in this section, and so we begin by recalling that definition from calculus/analysis. 1 n Definition 1.1. A sequence fxngn=1 is said to converge to a point x 2 R if for every > 0 there is a number N 2 N such that xn 2 B(x) for all n > N. 1 Remark 1.2. It is common to refer to the portion of a sequence fxngn=1 after some index 1 N|that is, the sequence fxngn=N+1|as a tail of the sequence. In this language, one would phrase the above definition as \for every > 0 there is a tail of the sequence inside B(x)." n Given what we have established about the topological space Rusual and its standard basis of -balls, we can see that this is equivalent to saying that there is a tail of the sequence inside any open set containing x; this is because the collection of -balls forms a basis for the usual topology, and thus given any open set U containing x there is an such that x 2 B(x) ⊆ U.
    [Show full text]
  • General Topology
    General Topology Tom Leinster 2014{15 Contents A Topological spaces2 A1 Review of metric spaces.......................2 A2 The definition of topological space.................8 A3 Metrics versus topologies....................... 13 A4 Continuous maps........................... 17 A5 When are two spaces homeomorphic?................ 22 A6 Topological properties........................ 26 A7 Bases................................. 28 A8 Closure and interior......................... 31 A9 Subspaces (new spaces from old, 1)................. 35 A10 Products (new spaces from old, 2)................. 39 A11 Quotients (new spaces from old, 3)................. 43 A12 Review of ChapterA......................... 48 B Compactness 51 B1 The definition of compactness.................... 51 B2 Closed bounded intervals are compact............... 55 B3 Compactness and subspaces..................... 56 B4 Compactness and products..................... 58 B5 The compact subsets of Rn ..................... 59 B6 Compactness and quotients (and images)............. 61 B7 Compact metric spaces........................ 64 C Connectedness 68 C1 The definition of connectedness................... 68 C2 Connected subsets of the real line.................. 72 C3 Path-connectedness.......................... 76 C4 Connected-components and path-components........... 80 1 Chapter A Topological spaces A1 Review of metric spaces For the lecture of Thursday, 18 September 2014 Almost everything in this section should have been covered in Honours Analysis, with the possible exception of some of the examples. For that reason, this lecture is longer than usual. Definition A1.1 Let X be a set. A metric on X is a function d: X × X ! [0; 1) with the following three properties: • d(x; y) = 0 () x = y, for x; y 2 X; • d(x; y) + d(y; z) ≥ d(x; z) for all x; y; z 2 X (triangle inequality); • d(x; y) = d(y; x) for all x; y 2 X (symmetry).
    [Show full text]
  • DEFINITIONS and THEOREMS in GENERAL TOPOLOGY 1. Basic
    DEFINITIONS AND THEOREMS IN GENERAL TOPOLOGY 1. Basic definitions. A topology on a set X is defined by a family O of subsets of X, the open sets of the topology, satisfying the axioms: (i) ; and X are in O; (ii) the intersection of finitely many sets in O is in O; (iii) arbitrary unions of sets in O are in O. Alternatively, a topology may be defined by the neighborhoods U(p) of an arbitrary point p 2 X, where p 2 U(p) and, in addition: (i) If U1;U2 are neighborhoods of p, there exists U3 neighborhood of p, such that U3 ⊂ U1 \ U2; (ii) If U is a neighborhood of p and q 2 U, there exists a neighborhood V of q so that V ⊂ U. A topology is Hausdorff if any distinct points p 6= q admit disjoint neigh- borhoods. This is almost always assumed. A set C ⊂ X is closed if its complement is open. The closure A¯ of a set A ⊂ X is the intersection of all closed sets containing X. A subset A ⊂ X is dense in X if A¯ = X. A point x 2 X is a cluster point of a subset A ⊂ X if any neighborhood of x contains a point of A distinct from x. If A0 denotes the set of cluster points, then A¯ = A [ A0: A map f : X ! Y of topological spaces is continuous at p 2 X if for any open neighborhood V ⊂ Y of f(p), there exists an open neighborhood U ⊂ X of p so that f(U) ⊂ V .
    [Show full text]
  • CLOSURES of EXPONENTIAL FAMILIES 3 the (Common) Convex Support of the P.M.’S in E Has Positive Probability Under These P.M.’S, by [7], Remark 3
    The Annals of Probability 2005, Vol. 33, No. 2, 582–600 DOI: 10.1214/009117904000000766 c Institute of Mathematical Statistics, 2005 CLOSURES OF EXPONENTIAL FAMILIES1 By Imre Csiszar´ and Frantiˇsek Matu´ˇs Hungarian Academy of Sciences and Academy of Sciences of the Czech Republic The variation distance closure of an exponential family with a convex set of canonical parameters is described, assuming no regu- larity conditions. The tools are the concepts of convex core of a mea- sure and extension of an exponential family, introduced previously by the authors, and a new concept of accessible faces of a convex set. Two other closures related to the information divergence are also characterized. 1. Introduction. Exponential families of probability measures (p.m.’s) include many of the parametric families frequently used in statistics, proba- bility and information theory. Their mathematical theory has been worked out to a considerable extent [1, 2, 3, 11]. Although limiting considerations are important and do appear in the literature, less attention has been paid to determining closures of exponential families. For families supported by a finite or countable set, closures were consid- ered in [1], pages 154–156, and [2], pages 191–201, respectively, the latter with regularity conditions. In the general case, different closure concepts come into account. Our main result, Theorem 2 in Section 3, determines the closure in variation distance (variation closure) of a full exponential fam- ily and, more generally, of any subfamily with a convex set of canonical parameters. Weak closures appear much harder to describe in general, but Theorem 1 in Section 3 is a step in that direction.
    [Show full text]
  • On the Structure of a Complete Metric Space
    Math 541 Lecture #3 I.13: Metric Spaces, Part II I.16: On the Structure of a Complete Metric Space 13.1: Separation and axioms of countability. Metric spaces enjoys many nice properties. Proposition 13.2. A metric space fX; dg is separable (i.e., has a countable dense subset) if and only if it satisfies the second axiom of countability. Proof. If the metric topology satisfies the second axiom of countability, then by Propo- sition 4.2, it is separable. So suppose that fX; dg is separable. Then there is a countable dense subset A of X. The collection of balls centered at points of A having rational radii forms a countable base for the topology. 16: On the structure of complete metric spaces. For a topological space fX; Ug, c a subset E of X is nowhere dense in X if E = X − E is dense in X, i.e., if the closure of X − E is X. Proposition. Here are some basic facts about nowhere dense sets. (i) If E is nowhere dense in X, then E is nowhere dense. (ii) A closed set E is nowhere dense if and only if E does not contain any nonempty open set. (iii) If E is nowhere dense and O is nonempty and open, then O − E contains an nonempty open set. Proof. (i) If E is nowhere dense, then X − E is dense in X. Since the closure of the closed set E is itself, then X − E = X − E is dense, so E is nowhere dense. (ii) For a closed set E, suppose that E is nowhere dense, so X − E is dense in X.
    [Show full text]
  • 1.1.3 Reminder of Some Simple Topological Concepts Definition 1.1.17
    1. Preliminaries The Hausdorffcriterion could be paraphrased by saying that smaller neigh- borhoods make larger topologies. This is a very intuitive theorem, because the smaller the neighbourhoods are the easier it is for a set to contain neigh- bourhoods of all its points and so the more open sets there will be. Proof. Suppose τ τ . Fixed any point x X,letU (x). Then, since U ⇒ ⊆ ∈ ∈B is a neighbourhood of x in (X,τ), there exists O τ s.t. x O U.But ∈ ∈ ⊆ O τ implies by our assumption that O τ ,soU is also a neighbourhood ∈ ∈ of x in (X,τ ). Hence, by Definition 1.1.10 for (x), there exists V (x) B ∈B s.t. V U. ⊆ Conversely, let W τ. Then for each x W ,since (x) is a base of ⇐ ∈ ∈ B neighbourhoods w.r.t. τ,thereexistsU (x) such that x U W . Hence, ∈B ∈ ⊆ by assumption, there exists V (x)s.t.x V U W .ThenW τ . ∈B ∈ ⊆ ⊆ ∈ 1.1.3 Reminder of some simple topological concepts Definition 1.1.17. Given a topological space (X,τ) and a subset S of X,the subset or induced topology on S is defined by τ := S U U τ . That is, S { ∩ | ∈ } a subset of S is open in the subset topology if and only if it is the intersection of S with an open set in (X,τ). Alternatively, we can define the subspace topology for a subset S of X as the coarsest topology for which the inclusion map ι : S X is continuous.
    [Show full text]
  • On Theta-Continuity and Strong Theta-Continuity
    See discussions, stats, and author profiles for this publication at: https://www.researchgate.net/publication/262649243 On theta-continuity and strong theta-continuity Article in Applied Mathematics E - Notes · January 2003 CITATIONS READS 0 4 1 author: M. Saleh Birzeit University 52 PUBLICATIONS 239 CITATIONS SEE PROFILE Some of the authors of this publication are also working on these related projects: weak transitivity in devanys chaos View project All content following this page was uploaded by M. Saleh on 25 May 2016. The user has requested enhancement of the downloaded file. All in-text references underlined in blue are added to the original document and are linked to publications on ResearchGate, letting you access and read them immediately. Applied Mathematics E-Notes, 3(2003), 42-48 c ISSN 1607-2510 Available free at mirror sites of http://www.math.nthu.edu.tw/ amen/ ∼ On θ-Continuity And Strong θ-Continuity ∗ Mohammed Saleh† Received 3 March 2002 Abstract P. E. Long and D. A. Carnahan in [6] and Noiri in [10] studied several prop- erties of a.c.S, a.c.H and weak continuity. In this paper it is shown that results similar to those in the above mentioned papers still hold for θ-continuity and strong θ-continuity. Furthermore, several decomposition theorems of θ-continuity and strong θ-continuity are obtained. 1 Introduction The concepts of δ-closure, θ-closure, δ-interior and θ-interior operators were first in- troduced by Veliˇcko. These operators have since been studied intensively by many authors. The collection of all δ-open sets in a topological space (X, Γ) forms a topol- ogy Γs on X, called the semiregularization topology of Γ, weaker than Γ and the class of all regular open sets in Γ forms an open basis for Γs.
    [Show full text]
  • Math 396. Interior, Closure, and Boundary We Wish to Develop Some Basic Geometric Concepts in Metric Spaces Which Make Precise C
    Math 396. Interior, closure, and boundary We wish to develop some basic geometric concepts in metric spaces which make precise certain intuitive ideas centered on the themes of \interior" and \boundary" of a subset of a metric space. One warning must be given. Although there are a number of results proven in this handout, none of it is particularly deep. If you carefully study the proofs (which you should!), then you'll see that none of this requires going much beyond the basic definitions. We will certainly encounter some serious ideas and non-trivial proofs in due course, but at this point the central aim is to acquire some linguistic ability when discussing some basic geometric ideas in a metric space. Thus, the main goal is to familiarize ourselves with some very convenient geometric terminology in terms of which we can discuss more sophisticated ideas later on. 1. Interior and closure Let X be a metric space and A X a subset. We define the interior of A to be the set ⊆ int(A) = a A some Br (a) A; ra > 0 f 2 j a ⊆ g consisting of points for which A is a \neighborhood". We define the closure of A to be the set A = x X x = lim an; with an A for all n n f 2 j !1 2 g consisting of limits of sequences in A. In words, the interior consists of points in A for which all nearby points of X are also in A, whereas the closure allows for \points on the edge of A".
    [Show full text]
  • CLOSURE OPERATORS I Dikran DIKRANJAN Eraldo GIULI
    View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector Topology and its Applications 27 (1987) 129-143 129 North-Holland CLOSURE OPERATORS I Dikran DIKRANJAN Institute of Mathematics, Bulgarian Academy of Sciences, Sojia, Bulgaria Eraldo GIULI* Department of Pure and Applied Mathematics, University of L’Aquila, 67100-L’Aquila, Italy Received 8 June 1986 Revised 24 September 1986 Closure operators in an (E, M)-category X are introduced as concrete endofunctors of the comma category whose objects are the elements of M. Various kinds of closure operators are studied. There is a Galois equivalence between the conglomerate of idempotent and weakly hereditary closure operators of X and the conglomerate of subclasses of M which are part of a factorization system. There is a one-to-one correspondence between the class of regular closure operators and the class of strongly epireflective subcategories of X. Every closure operator admits an idempotent hull and a weakly hereditary core. Various examples of additive closure operators in Top are given. For abelian categories standard closure operators are considered. It is shown that there is a one-to-one correspondence between the class of standard closure operators and the class of preradicals. Idempotenf weakly hereditary, standard closure operators correspond to idempotent radicals ( = torsion theories). AMS (MOS) Subj. Class.: 18A32, 18B30, 18E40, 54A05, 54B30, 18A40, A-regular morphism weakly hereditary core sequential closure (pre)-radical Introduction Closure operators have always been one of the main concepts in topology. For example, Herrlich [21] characterized coreflections in the category Top of topological spaces by means of Kuratowski closure operators finer than the ordinary closure.
    [Show full text]
  • Completion and Closure Cahiers De Topologie Et Géométrie Différentielle Catégoriques, Tome 41, No 2 (2000), P
    CAHIERS DE TOPOLOGIE ET GÉOMÉTRIE DIFFÉRENTIELLE CATÉGORIQUES DAVID HOLGATE Completion and closure Cahiers de topologie et géométrie différentielle catégoriques, tome 41, no 2 (2000), p. 101-119 <http://www.numdam.org/item?id=CTGDC_2000__41_2_101_0> © Andrée C. Ehresmann et les auteurs, 2000, tous droits réservés. L’accès aux archives de la revue « Cahiers de topologie et géométrie différentielle catégoriques » implique l’accord avec les conditions générales d’utilisation (http://www.numdam.org/conditions). Toute utilisation commerciale ou impression systématique est constitutive d’une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright. Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques http://www.numdam.org/ CAHIERS DE TOPOLOGIE ET Volume XLI-2 (2000) GEOMETRIE DIFFERENTIELLE CATEGORIQUES COMPLETION AND CLOSURE by David HOLGATE RESUME. La fermeture (ou, de fagon synonyme la densite) a tou- jours jou6 un role important dans la th6orie des compl6tions. S’ap- puyant sur des id6es de Birkhoff, une fermeture est extraite de ma- niere canonique d’un processus de completion r6flexive dans une categorie. Cette fermeture caract6rise la compl6tude et la compl6- tion elle-meme. La fermeture n’a pas seulement de bonnes propri6t6s internes, mais c’est la plus grande parmi les fermetures qui d6crivent la completion. Le theoreme principal montre que, equivalent aux descriptions fermeture/densite naturelle d’une completion, est le simple fait mar- quant que les r6flecteurs de completion sont exactement ceux qui pr6servent les plongements. De tels r6flecteurs peuvent 6tre deduits de la fermeture elle-m6me. Le role de la pr6servation de la ferme- ture et du plongement jette alors une nouvelle lumi6re sur les exem- ples de completion.
    [Show full text]