Part IB — Metric and Topological Spaces

Total Page:16

File Type:pdf, Size:1020Kb

Part IB — Metric and Topological Spaces Part IB | Metric and Topological Spaces Based on lectures by J. Rasmussen Notes taken by Dexter Chua Easter 2015 These notes are not endorsed by the lecturers, and I have modified them (often significantly) after lectures. They are nowhere near accurate representations of what was actually lectured, and in particular, all errors are almost surely mine. Metrics Definition and examples. Limits and continuity. Open sets and neighbourhoods. Characterizing limits and continuity using neighbourhoods and open sets. [3] Topology Definition of a topology. Metric topologies. Further examples. Neighbourhoods, closed sets, convergence and continuity. Hausdorff spaces. Homeomorphisms. Topologi- cal and non-topological properties. Completeness. Subspace, quotient and product topologies. [3] Connectedness Definition using open sets and integer-valued functions. Examples, including inter- vals. Components. The continuous image of a connected space is connected. Path- connectedness. Path-connected spaces are connected but not conversely. Connected open sets in Euclidean space are path-connected. [3] Compactness Definition using open covers. Examples: finite sets and [0, 1]. Closed subsets of compact spaces are compact. Compact subsets of a Hausdorff space must be closed. The compact subsets of the real line. Continuous images of compact sets are compact. Quotient spaces. Continuous real-valued functions on a compact space are bounded and attain their bounds. The product of two compact spaces is compact. The compact subsets of Euclidean space. Sequential compactness. [3] 1 Contents IB Metric and Topological Spaces Contents 0 Introduction 3 1 Metric spaces 4 1.1 Definitions . .4 1.2 Examples of metric spaces . .6 1.3 Norms . .7 1.4 Open and closed subsets . 10 2 Topological spaces 15 2.1 Definitions . 15 2.2 Sequences . 18 2.3 Closed sets . 19 2.4 Closure and interior . 19 2.4.1 Closure . 19 2.4.2 Interior . 21 2.5 New topologies from old . 22 2.5.1 Subspace topology . 22 2.5.2 Product topology . 23 2.5.3 Quotient topology . 25 3 Connectivity 27 3.1 Connectivity . 27 3.2 Path connectivity . 29 3.2.1 Higher connectivity* . 30 3.3 Components . 30 3.3.1 Path components . 30 3.3.2 Connected components . 31 4 Compactness 34 4.1 Compactness . 35 4.2 Products and quotients . 38 4.2.1 Products . 38 4.2.2 Quotients . 39 4.3 Sequential compactness . 40 4.4 Completeness . 41 2 0 Introduction IB Metric and Topological Spaces 0 Introduction The course Metric and Topological Spaces is divided into two parts. The first is on metric spaces, and the second is on topological spaces (duh). In Analysis, we studied real numbers a lot. We defined many properties such as convergence of sequences and continuity of functions. For example, if (xn) is a sequence in R, xn ! x means (8" > 0)(9N)(8n > N) jxn − xj < ": Similarly, a function f is continuous at x0 if (8" > 0)(9δ > 0)(8x) jx − x0j < δ ) jf(x) − f(x0)j < ": However, the definition of convergence doesn't really rely on xn being real numbers, except when calculating values of jxn − xj. But what does jxn − xj really mean? It is the distance between xn and x. To define convergence, we don't really need notions like subtraction and absolute values. We simply need a (sensible) notion of distance between points. Given a set X, we can define a metric (\distance function") d : X × X ! R, where d(x; y) is the distance between the points x and y. Then we say a sequence (xn) in X converges to x if (8" > 0)(9N)(8n > N) d(xn; x) < ": Similarly, a function f : X ! X is continuous if (8" > 0)(9δ > 0)(8x) d(x; x0) < δ ) d(f(x); f(x0)) < ": Of course, we will need the metric d to satisfy certain conditions such as being non-negative, and we will explore these technical details in the first part of the course. As people studied metric spaces, it soon became evident that metrics are not really that useful a notion. Given a set X, it is possible to find many different metrics that completely agree on which sequences converge and what functions are continuous. It turns out that it is not the metric that determines (most of) the properties of the space. Instead, it is the open sets induced by the metric (intuitively, an open set is a subset of X without \boundary", like an open interval). Metrics that induce the same open sets will have almost identical properties (apart from the actual distance itself). The idea of a topological space is to just keep the notion of open sets and abandon metric spaces, and this turns out to be a really good idea. The second part of the course is the study of these topological spaces and defining a lot of interesting properties just in terms of open sets. 3 1 Metric spaces IB Metric and Topological Spaces 1 Metric spaces 1.1 Definitions As mentioned in the introduction, given a set X, it is often helpful to have a notion of distance between points. This distance function is known as the metric. Definition (Metric space). A metric space is a pair (X; dX ) where X is a set (the space) and dX is a function dX : X × X ! R (the metric) such that for all x; y; z, { dX (x; y) ≥ 0 (non-negativity) { dX (x; y) = 0 iff x = y (identity of indiscernibles) { dX (x; y) = dX (y; x) (symmetry) { dX (x; z) ≤ dX (x; y) + dX (y; z) (triangle inequality) We will have two quick examples of metrics before going into other important definitions. We will come to more examples afterwards. Example. { (Euclidean \usual" metric) Let X = Rn. Let v u n uX 2 d(v; w) = jv − wj = t (vi − wi) : i=1 This is the usual notion of distance we have in the Rn vector space. It is not difficult to show that this is indeed a metric (the fourth axiom follows from the Cauchy-Schwarz inequality). { (Discrete metric) Let X be a set, and ( 1 x 6= y dX (x; y) = 0 x = y To show this is indeed a metric, we have to show it satisfies all the axioms. The first three axioms are trivially satisfied. How about the fourth? We can prove this by exhaustion. Since the distance function can only return 0 or 1, d(x; z) can be 0 or 1, while d(x; y) + d(y; z) can be 0, 1 or 2. For the fourth axiom to fail, we must have RHS < LHS. This can only happen if the right hand side is 0. But for the right hand side to be 0, we must have x = y = z. So the left hand side is also 0. So the fourth axiom is always satisfied. Given a metric space (X; d), we can generate another metric space by picking out a subset of X and reusing the same metric. Definition (Metric subspace). Let (X; dX ) be a metric space, and Y ⊆ X. Then (Y; dY ) is a metric space, where dY (a; b) = dX (a; b), and said to be a subspace of X. 4 1 Metric spaces IB Metric and Topological Spaces Example. Sn = fv 2 Rn+1 : jvj = 1g, the n-dimensional sphere, is a subspace of Rn+1. Finally, as promised, we come to the definition of convergent sequences and continuous functions. Definition (Convergent sequences). Let (xn) be a sequence in a metric space (X; dX ). We say (xn) converges to x 2 X, written xn ! x, if d(xn; x) ! 0 (as a real sequence). Equivalently, (8" > 0)(9N)(8n > N) d(xn; x) < ": Example. k { Let (vn) be a sequence in R with the Euclidean metric. Write vn = 1 k 1 k k i i (vn; ··· ; vn), and v = (v ; ··· ; v ) 2 R . Then vn ! v iff (vn) ! v for all i. 1 { Let X have the discrete metric, and suppose xn ! x. Pick " = 2 . 1 Then there is some N such that d(xn; x) < 2 whenever n > N. But if 1 d(xn; x) < 2 , we must have d(xn; x) = 0. So xn = x. Hence if xn ! x, then eventually all xn are equal to x. Similar to what we did in Analysis, we can show that limits are unique (if exist). Proposition. If (X; d) is a metric space, (xn) is a sequence in X such that 0 0 xn ! x, xn ! x , then x = x . Proof. For any " > 0, we know that there exists N such that d(xn; x) < "=2 if 0 0 0 n > N. Similarly, there exists some N such that d(xn; x ) < "=2 if n > N . Hence if n > max(N; N 0), then 0 ≤ d(x; x0) 0 ≤ d(x; xn) + d(xn; x ) 0 = d(xn; x) + d(xn; x ) ≤ ": So 0 ≤ d(x; x0) ≤ " for all " > 0. So d(x; x0) = 0, and x = x0. Note that to prove the above proposition, we used all of the four axioms. In the first line, we used non-negativity to say 0 ≤ d(x; x0). In the second line, we used triangle inequality. In the third line, we used symmetry to swap d(x; xn) with d(xn; x). Finally, we used the identity of indiscernibles to conclude that x = x0. To define a continuous function, here, we opt to use the sequence definition. We will later show that this is indeed equivalent to the " − δ definition, as well as a few other more useful definitions. Definition (Continuous function). Let (X; dX ) and (Y; dY ) be metric spaces, and f : X ! Y .
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]
  • 07. Homeomorphisms and Embeddings
    07. Homeomorphisms and embeddings Homeomorphisms are the isomorphisms in the category of topological spaces and continuous functions. Definition. 1) A function f :(X; τ) ! (Y; σ) is called a homeomorphism between (X; τ) and (Y; σ) if f is bijective, continuous and the inverse function f −1 :(Y; σ) ! (X; τ) is continuous. 2) (X; τ) is called homeomorphic to (Y; σ) if there is a homeomorphism f : X ! Y . Remarks. 1) Being homeomorphic is apparently an equivalence relation on the class of all topological spaces. 2) It is a fundamental task in General Topology to decide whether two spaces are homeomorphic or not. 3) Homeomorphic spaces (X; τ) and (Y; σ) cannot be distinguished with respect to their topological structure because a homeomorphism f : X ! Y not only is a bijection between the elements of X and Y but also yields a bijection between the topologies via O 7! f(O). Hence a property whose definition relies only on set theoretic notions and the concept of an open set holds if and only if it holds in each homeomorphic space. Such properties are called topological properties. For example, ”first countable" is a topological property, whereas the notion of a bounded subset in R is not a topological property. R ! − t Example. The function f : ( 1; 1) with f(t) = 1+jtj is a −1 x homeomorphism, the inverse function is f (x) = 1−|xj . 1 − ! b−a a+b The function g :( 1; 1) (a; b) with g(x) = 2 x + 2 is a homeo- morphism. Therefore all open intervals in R are homeomorphic to each other and homeomorphic to R .
    [Show full text]
  • Foliations of Three Dimensional Manifolds∗
    Foliations of Three Dimensional Manifolds∗ M. H. Vartanian December 17, 2007 Abstract The theory of foliations began with a question by H. Hopf in the 1930's: \Does there exist on S3 a completely integrable vector field?” By Frobenius' Theorem, this is equivalent to: \Does there exist on S3 a codimension one foliation?" A decade later, G. Reeb answered affirmatively by exhibiting a C1 foliation of S3 consisting of a single compact leaf homeomorphic to the 2-torus with the other leaves homeomorphic to R2 accumulating asymptotically on the compact leaf. Reeb's work raised the basic question: \Does every C1 codimension one foliation of S3 have a compact leaf?" This was answered by S. Novikov with a much stronger statement, one of the deepest results of foliation theory: Every C2 codimension one foliation of a compact 3-dimensional manifold with finite fundamental group has a compact leaf. The basic ideas leading to Novikov's Theorem are surveyed here.1 1 Introduction Intuitively, a foliation is a partition of a manifold M into submanifolds A of the same dimension that stack up locally like the pages of a book. Perhaps the simplest 3 nontrivial example is the foliation of R nfOg by concentric spheres induced by the 3 submersion R nfOg 3 x 7! jjxjj 2 R. Abstracting the essential aspects of this example motivates the following general ∗Survey paper written for S.N. Simic's Math 213, \Advanced Differential Geometry", San Jos´eState University, Fall 2007. 1We follow here the general outline presented in [Ca]. For a more recent and much broader survey of foliation theory, see [To].
    [Show full text]
  • I-Sequential Topological Spaces∗
    Applied Mathematics E-Notes, 14(2014), 236-241 c ISSN 1607-2510 Available free at mirror sites of http://www.math.nthu.edu.tw/ amen/ -Sequential Topological Spaces I Sudip Kumar Pal y Received 10 June 2014 Abstract In this paper a new notion of topological spaces namely, I-sequential topo- logical spaces is introduced and investigated. This new space is a strictly weaker notion than the first countable space. Also I-sequential topological space is a quotient of a metric space. 1 Introduction The idea of convergence of real sequence have been extended to statistical convergence by [2, 14, 15] as follows: If N denotes the set of natural numbers and K N then Kn denotes the set k K : k n and Kn stands for the cardinality of the set Kn. The natural densityf of the2 subset≤ Kg is definedj j by Kn d(K) = lim j j, n n !1 provided the limit exists. A sequence xn n N of points in a metric space (X, ) is said to be statistically f g 2 convergent to l if for arbitrary " > 0, the set K(") = k N : (xk, l) " has natural density zero. A lot of investigation has been donef on2 this convergence≥ andg its topological consequences after initial works by [5, 13]. It is easy to check that the family Id = A N : d(A) = 0 forms a non-trivial f g admissible ideal of N (recall that I 2N is called an ideal if (i) A, B I implies A B I and (ii) A I,B A implies B I.
    [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]
  • 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]
  • Zuoqin Wang Time: March 25, 2021 the QUOTIENT TOPOLOGY 1. The
    Topology (H) Lecture 6 Lecturer: Zuoqin Wang Time: March 25, 2021 THE QUOTIENT TOPOLOGY 1. The quotient topology { The quotient topology. Last time we introduced several abstract methods to construct topologies on ab- stract spaces (which is widely used in point-set topology and analysis). Today we will introduce another way to construct topological spaces: the quotient topology. In fact the quotient topology is not a brand new method to construct topology. It is merely a simple special case of the co-induced topology that we introduced last time. However, since it is very concrete and \visible", it is widely used in geometry and algebraic topology. Here is the definition: Definition 1.1 (The quotient topology). (1) Let (X; TX ) be a topological space, Y be a set, and p : X ! Y be a surjective map. The co-induced topology on Y induced by the map p is called the quotient topology on Y . In other words, −1 a set V ⊂ Y is open if and only if p (V ) is open in (X; TX ). (2) A continuous surjective map p :(X; TX ) ! (Y; TY ) is called a quotient map, and Y is called the quotient space of X if TY coincides with the quotient topology on Y induced by p. (3) Given a quotient map p, we call p−1(y) the fiber of p over the point y 2 Y . Note: by definition, the composition of two quotient maps is again a quotient map. Here is a typical way to construct quotient maps/quotient topology: Start with a topological space (X; TX ), and define an equivalent relation ∼ on X.
    [Show full text]
  • Topological Properties
    CHAPTER 4 Topological properties 1. Connectedness • Definitions and examples • Basic properties • Connected components • Connected versus path connected, again 2. Compactness • Definition and first examples • Topological properties of compact spaces • Compactness of products, and compactness in Rn • Compactness and continuous functions • Embeddings of compact manifolds • Sequential compactness • More about the metric case 3. Local compactness and the one-point compactification • Local compactness • The one-point compactification 4. More exercises 63 64 4. TOPOLOGICAL PROPERTIES 1. Connectedness 1.1. Definitions and examples. Definition 4.1. We say that a topological space (X, T ) is connected if X cannot be written as the union of two disjoint non-empty opens U, V ⊂ X. We say that a topological space (X, T ) is path connected if for any x, y ∈ X, there exists a path γ connecting x and y, i.e. a continuous map γ : [0, 1] → X such that γ(0) = x, γ(1) = y. Given (X, T ), we say that a subset A ⊂ X is connected (or path connected) if A, together with the induced topology, is connected (path connected). As we shall soon see, path connectedness implies connectedness. This is good news since, unlike connectedness, path connectedness can be checked more directly (see the examples below). Example 4.2. (1) X = {0, 1} with the discrete topology is not connected. Indeed, U = {0}, V = {1} are disjoint non-empty opens (in X) whose union is X. (2) Similarly, X = [0, 1) ∪ [2, 3] is not connected (take U = [0, 1), V = [2, 3]). More generally, if X ⊂ R is connected, then X must be an interval.
    [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]
  • 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]
  • 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]