DOCSLIB.ORG

Expansion Growth of Foliations Annales De La Faculté Des Sciences De Toulouse 6E Série, Tome 2, No 1 (1993), P

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Expansion Growth of Foliations Annales De La Faculté Des Sciences De Toulouse 6E Série, Tome 2, No 1 (1993), P ANNALES DE LA FACULTÉ DES SCIENCES DE TOULOUSE SHINJI EGASHIRA Expansion growth of foliations Annales de la faculté des sciences de Toulouse 6e série, tome 2, no 1 (1993), p. 15-52 <http://www.numdam.org/item?id=AFST_1993_6_2_1_15_0> © Université Paul Sabatier, 1993, tous droits réservés. L’accès aux archives de la revue « Annales de la faculté des sciences de Toulouse » (http://picard.ups-tlse.fr/~annales/) implique l’accord avec les conditions générales d’utilisation (http://www.numdam.org/conditions). Toute utilisation commerciale ou impression systématique est constitu- tive 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/ Annales de la Faculte des Sciences de Toulouse Vol. II, n° 1, 1993 Expansion growth of foliations(*) SHINJI EGASHIRA(1) RESUME. - Nous definissons la croissance de 1’elargissement transverse des feuilletages qui peuvent ctre consider~s comme une croissance du car- dinal maximal des ensembles separes d’un pseudo-groupe d’holonomie. Nous prouvons que la croissance de 1’elargissement transverse est un in- variant topologique des feuilletages, et nous calculons la croissance de 1’elargissement transverse de quelques feuilletages typiques de co dimen- sion 1.. ABSTRACT. - We define the expansion growth of foliations which is, roughly speaking, growth of the maximum cardinality of separating sets with respect to a holonomy pseudogroup. We prove that the expansion growth is topological invariant for foliations and we compute the expansion growth of several typical foliations of codimension 1.. 0. Introduction The entropy of foliations is defined by Ghys, Langevin and Walczak Their definition was done by generalizing Bowen’s definition of the entropy of dynamical systems [B]. Let F be a codimension q foliation of class C~ on a compact manifold M. When we fix a finite foliation cover U of ( M, ~’), we obtain the holonomy pseudogroup ?~ of local homeomorphisms of Rq induced by 1.~. Then we can define an integer (n E > 0) to be the maximum cardinality of (n, c)-separating sets with respect to the holonomy pseudogroup ?~ .. sn ( ~ ) is monotone increasing on n and monotone decreasing on e. The entropy of the foliation ~’ is defined by the following formula: ~ * ~ Recu le 12 Novembre 1992 (1)) Department of Mathematical Sciences, University of Tokyo, 7-3-1, Hongo, Bunkyoku, Tokyo, 113 (Japan) - 15 - When we fix a sufficiently small positive real number we notice that the monotone increasing map with respect to n represents the expansion of the foliation. For example, if ~ is a Reeb component then the growth of with respect to n is the linear growth. If ~’ is a linear foliation on then the growth of is the constant growth. In this paper, we consider the growth type of sn (E) defined in the growth set which is an extension of the usual growth set ( cf. [HH2]) and we prove that the growth of depends only on (M, ~’). Therefore it becomes a topological invariant for foliations. Moreover we define an integer to be the minimum cardinality of (n, c)-spanning sets with respect to the holonomy pseudogroup ~C and we show that the growth of is equal to the growth of We call it the expansion growth of (M, ~’). The expansion growth of foliations produces a number of numerical topological invariants for foliations. We also show that many of them are non-trivial invariants. In section 1, we define the growth set which is an extansion of the usual growth set. The expansion growth of foliations is defined as an element of this growth set. In section 2, we define the expansion growth of a foliation and we describe several important properties of the expansion growth. In section 3, we compute the expansion growth of several typical codimension 1 foliations. We also construct foliations which have various expansion growths. 1. Growth In this section, we define the growth of an increasing sequence of incresing functions. Let Z be the set of nonnegative increasing functions on N: Let ~’ be the set of increasing sequences of I. We regard Z as a subset of Z by the map We define the growth type of an element of l. We define a preorder « in as follows. For ’ , (hk) k~IN G ’, In the definition, we note that B F_ IN is independent of j E IN and A > 0 may depends on j E IN . The preorder -~ induces an equivalence relation ^_. in I. We define e to be the set of equivalence classes in I: ~ is the set of all growth types of increasing sequences of increasing functions and has the partial order ~ induced by the preorder . The equivalence written and is called the class of j~IN G T is by j~IN ~ growth type of The equivalence class of g ~ I ~ is simply written by [g]. Let ~* be the set of such growth types: e is equal to the partial ordered set of all growths of monotone increasing functions in the usual sense (cf. [HH2]) and £ can be considered as an extension of it.. For example, the following relation is easily seen Here [0] (resp. [1]) is the growth of the constant function whose value is 0 (resp. 1). [en] E ~ is said to have exactly exponential growth. For k E IN U ~0~_, E ~ is said to have exactly polynomial growth of degree ~. E ~ is said to be quasi-exponential if Next we define the finite sum and the finite product of elements of ~. For E ~~ For example, the following relations are easily seen. 2. Expansion growth of foliations In this section, we define the expansion growth of a foliation on a compact manifold and describe several properties of it. Let ~’ be a C~ codimension q foliation on a compact (p + q)-dimensional manifold M. We assume that aM = 0, however the same discussion is applicable in the case where 0 provided ~’ is tangent or transverse to aM. A = U, is called a triple of foliation covers of (M, ~’~ if it satisfies the following conditions. (1) Ub, U and Us are open coverings of M. (2) Ub = (Ub~ ~~’) A ~ u = and _ , ~pi )}A 1. (3) = - (4) 03C6i 03C6bi |Ui and cpZ . (5) ~~(Ub) = = and = = x Bq1(oi) where o = (o, ... , 0) E IRp, oi (7i, ... , 7i) E IRq, and J3~(~) = {.r e IRP; ~! A;} G !R~. (6) If ut 0, then there exists a local homeomorphism of IRq such that ~b = o ~b on Ub where ~b = pr and pr : IRP x IRq --~ IRq is the projection to the second factor. We put Bi = C IRq. We call the set (~p,;~ 1 (B2 (o~ x ~z~~ (z E Bi~ a plaque of U (or a plaque of Let be the set of all plaques of Lt. Fix a metric d on M. We define the diameter of U to be the maximum = of the diameters of Ui (i 1, ..., , A~: We define and width(U,Ub) as follows: Here for K, K’ ç M, Let Leb(U) denote the Lebesgue number of U. a a Let x be point of M and let n be natural number. Let Pi ..., Pn be to an x x a plaques of U. (Pi, ..., , Pn ) is said be n-chain of Pu at if is point of and n Let be Pi Pl Ø. Ui1, ... foliation neighborhoods ofU. , ... , is said to be an n-chain of U at x if there exists an n-chain ... of Pu at x (uniquely determined) such that is a plaque of . When an n-chain of U at x is the n-chain Uil ... given, of at x with respect to ... , is often written by ... For xl, ..., E M, Let be the set of all n-chains of U at xl and put For K C M, let C be the set of Pn such that there exists ~ E K an at x. and n-chain , ... , Pn ) of Pu For plaques P and P’ of Ui, put Here Pb (resp. is the plaque of Ub containing P (resp. P’). For x, y E M, we define an n-chain of Here is the plaque of where ... , is ~u = . If then at x uniquely determined by , ... E y) ~, y~ = d(x, y). We remark that usually dn is not continuous. However, we can easily show the following lemma. LEMMA 2.1.2014 dn is lower semi-continuous. Let n be a natural number and let e be a positive number. For x, y E M, x and y are said to be (n, c, A)-separated if y~ > ~. Otherwise x and y are said to be (n, c, A)-close. Let K be a subset of M. S ç M is said to be an (n, e, .~4., K)-separating set if 5’ is a subset of K and for any x, y E S, x and y are (n, c, A)-separated. R C M is said to be an (n, ~, r4., K)-spanning set if for any x E K, there exists y E R such that x and yare (n, ~, A)-close. Since M is compact, S is a finite set ( c f. lemma 2.14 and theorem 2.1 S ~ . Put PROPOSITION 2.2 rn (~, K) and K) are monotone increasing on n E IN and monotone decreasing on E > 0.. If K C K’ then rn (e, K) rn (e, K’) and K) sn (E, K~~. Proof. - We only prove that Let S C K be an (n, ~, .~4., K)-separating set with the maximum cardinal- ity.
Load more

Recommended publications
  • 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]
  • What Are Lyapunov Exponents, and Why Are They Interesting?
    BULLETIN (New Series) OF THE AMERICAN MATHEMATICAL SOCIETY Volume 54, Number 1, January 2017, Pages 79–105 http://dx.doi.org/10.1090/bull/1552 Article electronically published on September 6, 2016 WHAT ARE LYAPUNOV EXPONENTS, AND WHY ARE THEY INTERESTING? AMIE WILKINSON Introduction At the 2014 International Congress of Mathematicians in Seoul, South Korea, Franco-Brazilian mathematician Artur Avila was awarded the Fields Medal for “his profound contributions to dynamical systems theory, which have changed the face of the field, using the powerful idea of renormalization as a unifying principle.”1 Although it is not explicitly mentioned in this citation, there is a second unify- ing concept in Avila’s work that is closely tied with renormalization: Lyapunov (or characteristic) exponents. Lyapunov exponents play a key role in three areas of Avila’s research: smooth ergodic theory, billiards and translation surfaces, and the spectral theory of 1-dimensional Schr¨odinger operators. Here we take the op- portunity to explore these areas and reveal some underlying themes connecting exponents, chaotic dynamics and renormalization. But first, what are Lyapunov exponents? Let’s begin by viewing them in one of their natural habitats: the iterated barycentric subdivision of a triangle. When the midpoint of each side of a triangle is connected to its opposite vertex by a line segment, the three resulting segments meet in a point in the interior of the triangle. The barycentric subdivision of a triangle is the collection of 6 smaller triangles determined by these segments and the edges of the original triangle: Figure 1. Barycentric subdivision. Received by the editors August 2, 2016.
    [Show full text]
  • Soft Somewhere Dense Sets on Soft Topological Spaces
    Commun. Korean Math. Soc. 33 (2018), No. 4, pp. 1341{1356 https://doi.org/10.4134/CKMS.c170378 pISSN: 1225-1763 / eISSN: 2234-3024 SOFT SOMEWHERE DENSE SETS ON SOFT TOPOLOGICAL SPACES Tareq M. Al-shami Abstract. The author devotes this paper to defining a new class of generalized soft open sets, namely soft somewhere dense sets and to in- vestigating its main features. With the help of examples, we illustrate the relationships between soft somewhere dense sets and some celebrated generalizations of soft open sets, and point out that the soft somewhere dense subsets of a soft hyperconnected space coincide with the non-null soft β-open sets. Also, we give an equivalent condition for the soft cs- dense sets and verify that every soft set is soft somewhere dense or soft cs-dense. We show that a collection of all soft somewhere dense subsets of a strongly soft hyperconnected space forms a soft filter on the universe set, and this collection with a non-null soft set form a soft topology on the universe set as well. Moreover, we derive some important results such as the property of being a soft somewhere dense set is a soft topological property and the finite product of soft somewhere dense sets is soft some- where dense. In the end, we point out that the number of soft somewhere dense subsets of infinite soft topological space is infinite, and we present some results which associate soft somewhere dense sets with some soft topological concepts such as soft compact spaces and soft subspaces.
    [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]
  • 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]
  • The Continuum Hypothesis and Its Relation to the Lusin Set
    THE CONTINUUM HYPOTHESIS AND ITS RELATION TO THE LUSIN SET CLIVE CHANG Abstract. In this paper, we prove that the Continuum Hypothesis is equiv- alent to the existence of a subset of R called a Lusin set and the property that every subset of R with cardinality < c is of first category. Additionally, we note an interesting consequence of the measure of a Lusin set, specifically that it has measure zero. We introduce the concepts of ordinals and cardinals, as well as discuss some basic point-set topology. Contents 1. Introduction 1 2. Ordinals 2 3. Cardinals and Countability 2 4. The Continuum Hypothesis and Aleph Numbers 2 5. The Topology on R 3 6. Meagre (First Category) and Fσ Sets 3 7. The existence of a Lusin set, assuming CH 4 8. The Lebesque Measure on R 4 9. Additional Property of the Lusin set 4 10. Lemma: CH is equivalent to R being representable as an increasing chain of countable sets 5 11. The Converse: Proof of CH from the existence of a Lusin set and a property of R 6 12. Closing Comments 6 Acknowledgments 6 References 6 1. Introduction Throughout much of the early and middle twentieth century, the Continuum Hypothesis (CH) served as one of the premier problems in the foundations of math- ematical set theory, attracting the attention of countless famous mathematicians, most notably, G¨odel,Cantor, Cohen, and Hilbert. The conjecture first advanced by Cantor in 1877, makes a claim about the relationship between the cardinality of the continuum (R) and the cardinality of the natural numbers (N), in relation to infinite set hierarchy.
    [Show full text]
  • Dense Sets on Bigeneralized Topological Spaces
    Int. Journal of Math. Analysis, Vol. 7, 2013, no. 21, 999 - 1003 HIKARI Ltd, www.m-hikari.com Dense Sets on Bigeneralized Topological Spaces Supunnee Sompong Department of Mathematics and Statistics Faculty of Science and Technology Sakon Nakhon Rajabhat University Sakon Nakhon 47000, Thailand s [email protected] Copyright c 2013 Supunnee Sompong. 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. Abstract The purpose of this paper is to introduce the concept and some fundamental properties of dense sets on bigeneralized topological spaces. Keywords: bigeneralized topological spaces, dense sets 1 Introduction In 2010, the notion of bigeneralized topological spaces and the concepts of weakly functions were introduced by C. Boonpok [1]. He also investigated some of their characterizations. In 2012, S. Sompong and B. Rodjanadid [2], introduced a definition of dense sets in biminimal structure spaces and study some fundamental of their properties. In 2013, S. Sompong and S. Muangchan [3, 4], introduced the definition of boundary and exterior set on bigeneralized topological spaces and the concepts of basic properties. In this paper, we introduce the concept of dense sets on bigeneralized topo- logical spaces and study some properties. 2 Preliminaries Let X be a nonempty set and g be a collection of subsets of X. Then g is called a generalized topology (briefly GT)onX if ∅∈g and if Gi ∈ g for i ∈ I = ∅ then G = i∈I Gi ∈ g. 1000 S. Sompong By (X, g), we denote a nonempty set X with a generalized topology g on X and it is called a generalized topological space (briefly GTS) on X.
    [Show full text]
  • Topological Vector Spaces
    Topological Vector Spaces Maria Infusino University of Konstanz Winter Semester 2015/2016 Contents 1 Preliminaries3 1.1 Topological spaces ......................... 3 1.1.1 The notion of topological space.............. 3 1.1.2 Comparison of topologies ................. 6 1.1.3 Reminder of some simple topological concepts...... 8 1.1.4 Mappings between topological spaces........... 11 1.1.5 Hausdorff spaces...................... 13 1.2 Linear mappings between vector spaces ............. 14 2 Topological Vector Spaces 17 2.1 Definition and main properties of a topological vector space . 17 2.2 Hausdorff topological vector spaces................ 24 2.3 Quotient topological vector spaces ................ 25 2.4 Continuous linear mappings between t.v.s............. 29 2.5 Completeness for t.v.s........................ 31 1 3 Finite dimensional topological vector spaces 43 3.1 Finite dimensional Hausdorff t.v.s................. 43 3.2 Connection between local compactness and finite dimensionality 46 4 Locally convex topological vector spaces 49 4.1 Definition by neighbourhoods................... 49 4.2 Connection to seminorms ..................... 54 4.3 Hausdorff locally convex t.v.s................... 64 4.4 The finest locally convex topology ................ 67 4.5 Direct limit topology on a countable dimensional t.v.s. 69 4.6 Continuity of linear mappings on locally convex spaces . 71 5 The Hahn-Banach Theorem and its applications 73 5.1 The Hahn-Banach Theorem.................... 73 5.2 Applications of Hahn-Banach theorem.............. 77 5.2.1 Separation of convex subsets of a real t.v.s. 78 5.2.2 Multivariate real moment problem............ 80 Chapter 1 Preliminaries 1.1 Topological spaces 1.1.1 The notion of topological space The topology on a set X is usually defined by specifying its open subsets of X.
    [Show full text]
  • 7560 THEOREMS with PROOFS Definition a Compactification of A
    7560 THEOREMS WITH PROOFS De¯nition A compacti¯cation of a completely regular space X is a compact T2- space ®X and a homeomorphic embedding ® : X ! ®X such that ®(X) is dense in ®X. Remark. To explain the separation assumptions in the above de¯nition: we will only be concerned with compacti¯cations which are Hausdor®. Note that only completely regular spaces can have a T2-compacti¯cation. Remark. A few of the concepts and results below are from ¯rst year topology, and are included here for completeness. De¯nition. Let X be a locally compact non-compact Hausdor® space. The one-point compacti¯cation !X of X is the space X [ f1g, where points of X have their usual neighborhoods, and a neighborhood of 1 has the form f1g [ (X n K) where K is a compact subset of X. In the framework of the previous de¯nition, the map ! : X ! !X is the identity map on X. Of course, X is open in its one-point compacti¯cation. We'll see that this is the case for any compacti¯cation of a locally compact X. Lemma 1. If X is a dense subset of a T2 space Z, and X is locally compact, then X is open in Z. Proof. If x 2 X, let Nx be a compact neighborhood x in X and Ux = int(Nx). Then Ux = X \ Vx for some Vx open in Z. Thus Ux is dense in Vx and so Vx ½ V x = U x ½ N x = Nx where closures are taken in Z.
    [Show full text]
  • Topological Vector Spaces and Continuous Linear Functionals
    CHAPTER III TOPOLOGICAL VECTOR SPACES AND CONTINUOUS LINEAR FUNCTIONALS The marvelous interaction between linearity and topology is introduced in this chapter. Although the most familiar examples of this interaction may be normed linear spaces, we have in mind here the more subtle, and perhaps more important, topological vector spaces whose topologies are defined as the weakest topologies making certain collections of functions continuous. See the examples in Exercises 3.8 and 3.9, and particularly the Schwartz space S discussed in Exercise 3.10. DEFINITION. A topological vector space is a real (or complex) vector space X on which there is a Hausdorff topology such that: (1) The map (x, y) → x+y is continuous from X×X into X. (Addition is continuous.) and (2) The map (t, x) → tx is continuous from R × X into X (or C × X into X). (Scalar multiplication is continuous.) We say that a topological vector space X is a real or complex topolog- ical vector space according to which field of scalars we are considering. A complex topological vector space is obviously also a real topological vector space. A metric d on a vector space X is called translation-invariant if d(x+ z, y + z) = d(x, y) for all x, y, z ∈ X. If the topology on a topological vector space X is determined by a translation-invariant metric d, we call X (or (X, d)) a metrizable vector space. If x is an element of a metrizable vector space (X, d), we denote by B(x) the ball of radius 43 44 CHAPTER III around x; i.e., B(x) = {y : d(x, y) < }.
    [Show full text]
  • Planetmath: Topological Space
    (more info) Math for the people, by the people. Encyclopedia | Requests | Forums | Docs | Wiki | Random | RSS Advanced search topological space (Definition) "topological space" is owned by djao. [ full author list (2) ] (more info) Math for the people, by the people. Encyclopedia | Requests | Forums | Docs | Wiki | Random | RSS Advanced search compact (Definition) "compact" is owned by djao. [ full author list (2) ] Dense set 1 Dense set In topology and related areas of mathematics, a subset A of a topological space X is called dense (in X) if any point x in X belongs to A or is a limit point of A.[1] Informally, for every point in X, the point is either in A or arbitrarily "close" to a member of A - for instance, every real number is either a rational number or has one arbitrarily close to it (see Diophantine approximation). Formally, a subset A of a topological space X is dense in X if for any point x in X, any neighborhood of x contains at least one point from A. Equivalently, A is dense in X if and only if the only closed subset of X containing A is X itself. This can also be expressed by saying that the closure of A is X, or that the interior of the complement of A is empty. The density of a topological space X is the least cardinality of a dense subset of X. Density in metric spaces An alternative definition of dense set in the case of metric spaces is the following. When the topology of X is given by a metric, the closure of A in X is the union of A and the set of all limits of sequences of elements in A (its limit points), Then A is dense in X if Note that .
    [Show full text]
  • Measure and Category
    Measure and Category Marianna Cs¨ornyei [email protected] http:/www.ucl.ac.uk/∼ucahmcs 1 / 96 A (very short) Introduction to Cardinals I The cardinality of a set A is equal to the cardinality of a set B, denoted |A| = |B|, if there exists a bijection from A to B. I A countable set A is an infinite set that has the same cardinality as the set of natural numbers N. That is, the elements of the set can be listed in a sequence A = {a1, a2, a3,... }. If an infinite set is not countable, we say it is uncountable. I The cardinality of the set of real numbers R is called continuum. 2 / 96 Examples of Countable Sets I The set of integers Z = {0, 1, −1, 2, −2, 3, −3,... } is countable. I The set of rationals Q is countable. For each positive integer k there are only a finite number of p rational numbers q in reduced form for which |p| + q = k. List those for which k = 1, then those for which k = 2, and so on: 0 1 −1 2 −2 1 −1 1 −1 = , , , , , , , , ,... Q 1 1 1 1 1 2 2 3 3 I Countable union of countable sets is countable. This follows from the fact that N can be decomposed as the union of countable many sequences: 1, 2, 4, 8, 16,... 3, 6, 12, 24,... 5, 10, 20, 40,... 7, 14, 28, 56,... 3 / 96 Cantor Theorem Theorem (Cantor) For any sequence of real numbers x1, x2, x3,..
    [Show full text]