DOCSLIB.ORG

Compact Metric Space

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Compact Metric Space Compact Metric Space Yongheng Zhang When we say a metric space X is compact, usually we mean any open covering of X has a finite subcovering. If we have a collection of open sets which covers X, no matter whether it is a covering of a subcovering, the collection of the complement of each open set has an empty intersection by De Morgen's law and vice versa. So another formulation of compactness is that any collection of closed subsets of X with an empty intersection has a finite subcollection also with an empty intersection. Furthermore, taking the contrapositive we have an equivalent defintion of compactness: for any col- lection C of closed subsets of X, if every finite subcollection of C has nonempty intersection, then the intersection of the sets in C is nonempty. We say a collection of sets of X has the finite intersection property if every finite subcollection of X has nonempty intersetion. So a concise version of the previous statement is A space X is compact if and only if every collection of closed subsets of X with the finite intersec- tion property has nonempty intersection. Equipped with the above statement, we will find a fine connection between compactness and the Bolzano-Weierstrass property. Recall that when we say a space X has the Bolzano-Weierstrass prop- erty, we mean any sequence in X has a cluster point (Is the cluster point in X?). In fact, compactness implies the Bolzano-Weierstrass property. To see why, let < xn > be a sequence in a 1 1 compact space X. Then we consider the collection of closed sets A = fAkgk=1 where Ak = fxngn=k. Why is Ak 2 X? Clearly A has the finite intersection property. Because X is compact, we know T1 that k=1 Ak is nonempty. Let x be in the intersection. Let Ox be an open set containing x. For any N, x 2 AN = fxN ; xN+1;:::g. So there is n ≥ N such that xn 2 Ox. This precisely says that x is a cluster point of < xn >. Recall that a space X is called sequentially compact if every sequence in it has a convergent sub- sequence. It can be readily checked that sequential compactness is the same as having the Bolzano- Weierstrass property (In the backward direction, we construct a subsequence converging to the cluster point). Now assume only that X is sequentially compact. Let f 2 C(X). We will see that f assumes its minimum (and maximum) and use it to show the existence of Lebesgue number of a compact metric space. Let m = infx2X ff(x)g. By successively shrinking > 0 in the defintion of infimum, we obtain a sequence < xn > in X such that limn!1 f(xn) = m. Since X is sequentially compact, < xn > has a subsequence < xnk > converging to some point x 2 X. Because f is continuous, we have f(x) = limk!1 f(xnk ). Thus, m = f(x), as desired. Similarly the maximum is achieved. Given an open covering U of a metric space X, if there is a number > 0 such that for any 0 < δ < and x 2 X, there is O 2 U such that the open ball Bδ(x) is contained in O, then we say that is the Lebesgue number of the open covering U. In fact, if a space X is sequentially compact, then its Lebesgue number exists. Now we prove this. Let U be an open covering of sequentially compact X. If X 2 U, then ... On the other hand, suppose X2 = U. Define a function f by f(x) = supfr : there is O 2 U such that Br(x) ⊂ 1 2 Og. Obviously, f > 0. (But why f < 1?) Notice that for any x; y 2 X, f(y) ≥ f(x) − ρ(x; y). Thus, f is continuous. By sequential compactness, f assumes its minimum, i.e., there is x 2 X and > 0 such that infx2X f(x) = > 0. Therefore, for any δ < and x 2 X, there is O 2 U such that Bδ(x) ⊂ O. So is the Lebesgue number. (if we increase , then we can not guarantee the uniform containment.) A metric space X is called totally bounded if for any > 0 it can be covered by a finite number of open balls with radius . One of its key words is finiteness, so it is closely related to compactness. Later we will use it to show that being sequentially compact implies compactness. But first, we prove that a sequentially compact space is totally bounded. We prove the contrapositive. Suppose X is not totally bounded, i.e., there is > 0 such that for any finite collection of points fx1; : : : ; xN g in X, there is a point x 2 X such that ρ(xn; x) ≥ for all n = 1;:::;N. Given this > 0, there is x1 2 X such that there is x2 2 X such that ρ(x2; x1) ≥ . Then, there is x3 2 X such that ρ(x3; xn) ≥ for n = 1; 2. Inductively, there is xN+1 2 X such that ρ(xN+1; xn) ≥ for all n = 1;:::;N. Hence, we have a sequence < xn > such that any two terms in them are at a distance at least . Therefore, it is impossible for < xn > to have a convergent subsequence. Given the existence of the Lebesgue number and the property of totally bounded of a sequentially compact space, we show that the space is compact. To see this, let U be an open covering of compact X; let > 0 be the Lebesgue number and let 0 < δ < . Because X is totally bounded, there is a N finite collection of points fx1; : : : ; xN g in X such that fBδ(xn)gn=1 covers X. But each Bδ(xn) is in N some open set Oxn 2 U. Consequently, the finite collection fOxn gn=1 covers X and we are done..
Load more

Recommended publications
  • Metric Geometry in a Tame Setting
    University of California Los Angeles Metric Geometry in a Tame Setting A dissertation submitted in partial satisfaction of the requirements for the degree Doctor of Philosophy in Mathematics by Erik Walsberg 2015 c Copyright by Erik Walsberg 2015 Abstract of the Dissertation Metric Geometry in a Tame Setting by Erik Walsberg Doctor of Philosophy in Mathematics University of California, Los Angeles, 2015 Professor Matthias J. Aschenbrenner, Chair We prove basic results about the topology and metric geometry of metric spaces which are definable in o-minimal expansions of ordered fields. ii The dissertation of Erik Walsberg is approved. Yiannis N. Moschovakis Chandrashekhar Khare David Kaplan Matthias J. Aschenbrenner, Committee Chair University of California, Los Angeles 2015 iii To Sam. iv Table of Contents 1 Introduction :::::::::::::::::::::::::::::::::::::: 1 2 Conventions :::::::::::::::::::::::::::::::::::::: 5 3 Metric Geometry ::::::::::::::::::::::::::::::::::: 7 3.1 Metric Spaces . 7 3.2 Maps Between Metric Spaces . 8 3.3 Covers and Packing Inequalities . 9 3.3.1 The 5r-covering Lemma . 9 3.3.2 Doubling Metrics . 10 3.4 Hausdorff Measures and Dimension . 11 3.4.1 Hausdorff Measures . 11 3.4.2 Hausdorff Dimension . 13 3.5 Topological Dimension . 15 3.6 Left-Invariant Metrics on Groups . 15 3.7 Reductions, Ultralimits and Limits of Metric Spaces . 16 3.7.1 Reductions of Λ-valued Metric Spaces . 16 3.7.2 Ultralimits . 17 3.7.3 GH-Convergence and GH-Ultralimits . 18 3.7.4 Asymptotic Cones . 19 3.7.5 Tangent Cones . 22 3.7.6 Conical Metric Spaces . 22 3.8 Normed Spaces . 23 4 T-Convexity :::::::::::::::::::::::::::::::::::::: 24 4.1 T-convex Structures .
    [Show full text]
  • K-THEORY of FUNCTION RINGS Theorem 7.3. If X Is A
    View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector Journal of Pure and Applied Algebra 69 (1990) 33-50 33 North-Holland K-THEORY OF FUNCTION RINGS Thomas FISCHER Johannes Gutenberg-Universitit Mainz, Saarstrage 21, D-6500 Mainz, FRG Communicated by C.A. Weibel Received 15 December 1987 Revised 9 November 1989 The ring R of continuous functions on a compact topological space X with values in IR or 0Z is considered. It is shown that the algebraic K-theory of such rings with coefficients in iZ/kH, k any positive integer, agrees with the topological K-theory of the underlying space X with the same coefficient rings. The proof is based on the result that the map from R6 (R with discrete topology) to R (R with compact-open topology) induces a natural isomorphism between the homologies with coefficients in Z/kh of the classifying spaces of the respective infinite general linear groups. Some remarks on the situation with X not compact are added. 0. Introduction For a topological space X, let C(X, C) be the ring of continuous complex valued functions on X, C(X, IR) the ring of continuous real valued functions on X. KU*(X) is the topological complex K-theory of X, KO*(X) the topological real K-theory of X. The main goal of this paper is the following result: Theorem 7.3. If X is a compact space, k any positive integer, then there are natural isomorphisms: K,(C(X, C), Z/kZ) = KU-‘(X, Z’/kZ), K;(C(X, lR), Z/kZ) = KO -‘(X, UkZ) for all i 2 0.
    [Show full text]
  • MTH 304: General Topology Semester 2, 2017-2018
    MTH 304: General Topology Semester 2, 2017-2018 Dr. Prahlad Vaidyanathan Contents I. Continuous Functions3 1. First Definitions................................3 2. Open Sets...................................4 3. Continuity by Open Sets...........................6 II. Topological Spaces8 1. Definition and Examples...........................8 2. Metric Spaces................................. 11 3. Basis for a topology.............................. 16 4. The Product Topology on X × Y ...................... 18 Q 5. The Product Topology on Xα ....................... 20 6. Closed Sets.................................. 22 7. Continuous Functions............................. 27 8. The Quotient Topology............................ 30 III.Properties of Topological Spaces 36 1. The Hausdorff property............................ 36 2. Connectedness................................. 37 3. Path Connectedness............................. 41 4. Local Connectedness............................. 44 5. Compactness................................. 46 6. Compact Subsets of Rn ............................ 50 7. Continuous Functions on Compact Sets................... 52 8. Compactness in Metric Spaces........................ 56 9. Local Compactness.............................. 59 IV.Separation Axioms 62 1. Regular Spaces................................ 62 2. Normal Spaces................................ 64 3. Tietze's extension Theorem......................... 67 4. Urysohn Metrization Theorem........................ 71 5. Imbedding of Manifolds..........................
    [Show full text]
  • Compact Non-Hausdorff Manifolds
    A.G.M. Hommelberg Compact non-Hausdorff Manifolds Bachelor Thesis Supervisor: dr. D. Holmes Date Bachelor Exam: June 5th, 2014 Mathematisch Instituut, Universiteit Leiden 1 Contents 1 Introduction 3 2 The First Question: \Can every compact manifold be obtained by glueing together a finite number of compact Hausdorff manifolds?" 4 2.1 Construction of (Z; TZ ).................................5 2.2 The space (Z; TZ ) is a Compact Manifold . .6 2.3 A Negative Answer to our Question . .8 3 The Second Question: \If (R; TR) is a compact manifold, is there always a surjective ´etale map from a compact Hausdorff manifold to R?" 10 3.1 Construction of (R; TR)................................. 10 3.2 Etale´ maps and Covering Spaces . 12 3.3 A Negative Answer to our Question . 14 4 The Fundamental Groups of (R; TR) and (Z; TZ ) 16 4.1 Seifert - Van Kampen's Theorem for Fundamental Groupoids . 16 4.2 The Fundamental Group of (R; TR)........................... 19 4.3 The Fundamental Group of (Z; TZ )........................... 23 2 Chapter 1 Introduction One of the simplest examples of a compact manifold (definitions 2.0.1 and 2.0.2) that is not Hausdorff (definition 2.0.3), is the space X obtained by glueing two circles together in all but one point. Since the circle is a compact Hausdorff manifold, it is easy to prove that X is indeed a compact manifold. It is also easy to show X is not Hausdorff. Many other examples of compact manifolds that are not Hausdorff can also be created by glueing together compact Hausdorff manifolds.
    [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]
  • Classification of Compact 2-Manifolds
    Virginia Commonwealth University VCU Scholars Compass Theses and Dissertations Graduate School 2016 Classification of Compact 2-manifolds George H. Winslow Virginia Commonwealth University Follow this and additional works at: https://scholarscompass.vcu.edu/etd Part of the Geometry and Topology Commons © The Author Downloaded from https://scholarscompass.vcu.edu/etd/4291 This Thesis is brought to you for free and open access by the Graduate School at VCU Scholars Compass. It has been accepted for inclusion in Theses and Dissertations by an authorized administrator of VCU Scholars Compass. For more information, please contact [email protected]. Abstract Classification of Compact 2-manifolds George Winslow It is said that a topologist is a mathematician who can not tell the difference between a doughnut and a coffee cup. The surfaces of the two objects, viewed as topological spaces, are homeomorphic to each other, which is to say that they are topologically equivalent. In this thesis, we acknowledge some of the most well-known examples of surfaces: the sphere, the torus, and the projective plane. We then ob- serve that all surfaces are, in fact, homeomorphic to either the sphere, the torus, a connected sum of tori, a projective plane, or a connected sum of projective planes. Finally, we delve into algebraic topology to determine that the aforementioned sur- faces are not homeomorphic to one another, and thus we can place each surface into exactly one of these equivalence classes. Thesis Director: Dr. Marco Aldi Classification of Compact 2-manifolds by George Winslow Bachelor of Science University of Mary Washington Submitted in Partial Fulfillment of the Requirements for the Degree of Master of Science in the Department of Mathematics Virginia Commonwealth University 2016 Dedication For Lily ii Abstract It is said that a topologist is a mathematician who can not tell the difference between a doughnut and a coffee cup.
    [Show full text]
  • Discrete Geometric Homotopy Theory and Critical Values of Metric Spaces Leonard Duane Wilkins [email protected]
    View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by University of Tennessee, Knoxville: Trace University of Tennessee, Knoxville Trace: Tennessee Research and Creative Exchange Doctoral Dissertations Graduate School 5-2011 Discrete Geometric Homotopy Theory and Critical Values of Metric Spaces Leonard Duane Wilkins [email protected] Recommended Citation Wilkins, Leonard Duane, "Discrete Geometric Homotopy Theory and Critical Values of Metric Spaces. " PhD diss., University of Tennessee, 2011. https://trace.tennessee.edu/utk_graddiss/1039 This Dissertation is brought to you for free and open access by the Graduate School at Trace: Tennessee Research and Creative Exchange. It has been accepted for inclusion in Doctoral Dissertations by an authorized administrator of Trace: Tennessee Research and Creative Exchange. For more information, please contact [email protected]. To the Graduate Council: I am submitting herewith a dissertation written by Leonard Duane Wilkins entitled "Discrete Geometric Homotopy Theory and Critical Values of Metric Spaces." I have examined the final electronic copy of this dissertation for form and content and recommend that it be accepted in partial fulfillment of the requirements for the degree of Doctor of Philosophy, with a major in Mathematics. Conrad P. Plaut, Major Professor We have read this dissertation and recommend its acceptance: James Conant, Fernando Schwartz, Michael Guidry Accepted for the Council: Dixie L. Thompson Vice Provost and Dean of the Graduate School (Original signatures are on file with official student records.) To the Graduate Council: I am submitting herewith a dissertation written by Leonard Duane Wilkins entitled \Discrete Geometric Homotopy Theory and Critical Values of Metric Spaces." I have examined the final electronic copy of this dissertation for form and content and recommend that it be accepted in partial fulfillment of the requirements for the degree of Doctor of Philosophy, with a major in Mathematics.
    [Show full text]
  • Notes on Metric Spaces
    Notes on Metric Spaces These notes introduce the concept of a metric space, which will be an essential notion throughout this course and in others that follow. Some of this material is contained in optional sections of the book, but I will assume none of that and start from scratch. Still, you should check the corresponding sections in the book for a possibly different point of view on a few things. The main idea to have in mind is that a metric space is some kind of generalization of R in the sense that it is some kind of \space" which has a notion of \distance". Having such a \distance" function will allow us to phrase many concepts from real analysis|such as the notions of convergence and continuity|in a more general setting, which (somewhat) surprisingly makes many things actually easier to understand. Metric Spaces Definition 1. A metric on a set X is a function d : X × X ! R such that • d(x; y) ≥ 0 for all x; y 2 X; moreover, d(x; y) = 0 if and only if x = y, • d(x; y) = d(y; x) for all x; y 2 X, and • d(x; y) ≤ d(x; z) + d(z; y) for all x; y; z 2 X. A metric space is a set X together with a metric d on it, and we will use the notation (X; d) for a metric space. Often, if the metric d is clear from context, we will simply denote the metric space (X; d) by X itself.
    [Show full text]
  • Compact and Totally Bounded Metric Spaces
    Appendix A Compact and Totally Bounded Metric Spaces Definition A.0.1 (Diameter of a set) The diameter of a subset E in a metric space (X, d) is the quantity diam(E) = sup d(x, y). x,y∈E ♦ Definition A.0.2 (Bounded and totally bounded sets)A subset E in a metric space (X, d) is said to be 1. bounded, if there exists a number M such that E ⊆ B(x, M) for every x ∈ E, where B(x, M) is the ball centred at x of radius M. Equivalently, d(x1, x2) ≤ M for every x1, x2 ∈ E and some M ∈ R; > { ,..., } 2. totally bounded, if for any 0 there exists a finite subset x1 xn of X ( , ) = ,..., = such that the (open or closed) balls B xi , i 1 n cover E, i.e. E n ( , ) i=1 B xi . ♦ Clearly a set is bounded if its diameter is finite. We will see later that a bounded set may not be totally bounded. Now we prove the converse is always true. Proposition A.0.1 (Total boundedness implies boundedness) In a metric space (X, d) any totally bounded set E is bounded. Proof Total boundedness means we may choose = 1 and find A ={x1,...,xn} in X such that for every x ∈ E © The Editor(s) (if applicable) and The Author(s), under exclusive 429 license to Springer Nature Switzerland AG 2020 S. Gentili, Measure, Integration and a Primer on Probability Theory, UNITEXT 125, https://doi.org/10.1007/978-3-030-54940-4 430 Appendix A: Compact and Totally Bounded Metric Spaces inf d(xi , x) ≤ 1.
    [Show full text]
  • 16. Compactness
    16. Compactness 1 Motivation While metrizability is the analyst's favourite topological property, compactness is surely the topologist's favourite topological property. Metric spaces have many nice properties, like being first countable, very separative, and so on, but compact spaces facilitate easy proofs. They allow you to do all the proofs you wished you could do, but never could. The definition of compactness, which we will see shortly, is quite innocuous looking. What compactness does for us is allow us to turn infinite collections of open sets into finite collections of open sets that do essentially the same thing. Compact spaces can be very large, as we will see in the next section, but in a strong sense every compact space acts like a finite space. This behaviour allows us to do a lot of hands-on, constructive proofs in compact spaces. For example, we can often take maxima and minima where in a non-compact space we would have to take suprema and infima. We will be able to intersect \all the open sets" in certain situations and end up with an open set, because finitely many open sets capture all the information in the whole collection. We will specifically prove an important result from analysis called the Heine-Borel theorem n that characterizes the compact subsets of R . This result is so fundamental to early analysis courses that it is often given as the definition of compactness in that context. 2 Basic definitions and examples Compactness is defined in terms of open covers, which we have talked about before in the context of bases but which we formally define here.
    [Show full text]
  • Magnitude Homology
    Magnitude homology Tom Leinster Edinburgh A theme of this conference so far When introducing a piece of category theory during a talk: 1. apologize; 2. blame John Baez. A theme of this conference so far When introducing a piece of category theory during a talk: 1. apologize; 2. blame John Baez. 1. apologize; 2. blame John Baez. A theme of this conference so far When introducing a piece of category theory during a talk: Plan 1. The idea of magnitude 2. The magnitude of a metric space 3. The idea of magnitude homology 4. The magnitude homology of a metric space 1. The idea of magnitude Size For many types of mathematical object, there is a canonical notion of size. • Sets have cardinality. It satisfies jX [ Y j = jX j + jY j − jX \ Y j jX × Y j = jX j × jY j : n • Subsets of R have volume. It satisfies vol(X [ Y ) = vol(X ) + vol(Y ) − vol(X \ Y ) vol(X × Y ) = vol(X ) × vol(Y ): • Topological spaces have Euler characteristic. It satisfies χ(X [ Y ) = χ(X ) + χ(Y ) − χ(X \ Y ) (under hypotheses) χ(X × Y ) = χ(X ) × χ(Y ): Challenge Find a general definition of `size', including these and other examples. One answer The magnitude of an enriched category. Enriched categories A monoidal category is a category V equipped with a product operation. A category X enriched in V is like an ordinary category, but each HomX(X ; Y ) is now an object of V (instead of a set). linear categories metric spaces categories enriched posets categories monoidal • Vect • Set • ([0; 1]; ≥) categories • (0 ! 1) The magnitude of an enriched category There is a general definition of the magnitude jXj of an enriched category.
    [Show full text]