DOCSLIB.ORG

Chapter 9 the Topology of Metric Spaces

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Chapter 9 the Topology of Metric Spaces Chapter 9 The Topology of Metric Spaces 9.1 Abstract Topological Spaces We undertake a study of metric spaces because we wish to study, among other things the set of continuous functions defined on Rn,andRn is a simple instance of a metric space, which we shall shortly define. The topology of a space is of particular interest to us, because the topology of the space and the set of continuous function defined on that space are intimately connected, as we shall soon see. Definition 9.1 Let X be a set, and C a collection of subsets of X.Then(X, C) is called a topological space, and the elements of C are called the open sets of X, provided the following hold: 1. ∅∈C. 2. X ∈C. 3. If X, Y ∈Cthen X ∩ Y ∈C. 4. The union of any number (i.e. finite or infinite number) of elements of C is again an element of C. 1 2 CHAPTER 9. THE TOPOLOGY OF METRIC SPACES Note that by induction, (3) implies that the intersection of any finite number of elements of C is an element of C,orsaidanotherway:C is closed under finite intersection, and from (4), C is closed under arbitrary union. Often, having exhibited the topological space (X, C), we will often refer to “an open set O in the topological space X”, understanding that that means O ∈C. Example 1 Let X = {0, 1}, that is, a set consisting of two elements. Then if we let C = {{0}, {1},X,∅},then(X, C) is a topological space. This is true because (1) and (2) can be verified by inspection. (3) and (4) require that certain subsets of X are elements of C, but we have chosen C to be all subsets of C, which make (3) and (4) hold automatically. This reasoning generalizes to the following example: Example 2 Let X be an arbitrary set, and C be the set of all subsets of X, including both ∅ and X.ThenaccordingtothereasoningofExample1,(X, C) is a topological space. This is called the discrete topology for X. Example 3 Let X be arbitrary, and let C = {∅,X}.Then(X, C) is a topological space, and the topology is called the trivial topology. Example 4 [The Usual Topology for R1.] Let X =(−∞, ∞), and let C consist of all intervals of the form (a, b), the arbitrary union of such intervals, and the intersection of any finite number of elements of C.Then(X, C) is a topological space, and the open sets are just the open sets we studied in Chapter 1. To see this, first note that since (1, 2) ∈Cand (3, 4) ∈C, ∅ =(1, 2) ∩ (3, 4) ∈C. Further, (x − 1,x+1)∈C x∈X which implies that X ∈C. To prove (3), suppose (a, b)and(c, d) are elements of C.Thenif (a, b) ∩ (c, d) = ∅,either(a, b) ⊂ (c, d)or(c, d) ⊂ (a, b)ora<c<b<dor c<a<d<b. In each of these four cases the intersection is an interval.1 The intersection of two arbitrary 1In Exercise 1 you are asked to compute the intersection explicitly, for each of the four cases. 9.1. ABSTRACT TOPOLOGICAL SPACES 3 unions of intervals is again a union of such intervals, and hence so is the intersection of finitely many such unions. Example 5 [The Right Order Topology] Let X =(−∞, ∞), and let C consist of all intervals of the form [a, ∞), arbitrary unions of such intervals, the empty set, and X.Then (X, C) is a topological space. We conclude the introduction with a simple and yet powerful theorem about open sets: Theorem 9.1 Let (X, C) be a topological space. Then O ⊂ X is an open set, that is, O ∈C, if and only if for every x ∈ O there exists a B ∈Csuch that x ∈ B ⊂ O. Proof: If O ∈C, then for every x ∈ O, there exists a B ∈C(O will play the role of B)so the theorem is trivially true. Conversely, if for every x ∈ O there exists a B ∈Csuch that x ∈ B ⊂ O, then indicating the dependence of B upon x by Bx, O ⊂ Bx ⊂ O {x:x∈O} and therefore O = Bx. {x:x∈O} Since arbitrary unions of sets in C are again in C, it follows that O ∈C. Continuous Functions on an Arbitrary Topological Space Definition 9.2 Let (X, C) and (Y,C) be two topological spaces. Suppose f is a function whose domain is X and whose range is contained in Y .Thenf is continuous if and only if the following condition is met: For every open set O in the topological space (Y,C),thesetf −1(O) is open in the topo- logical space (X, C). 4 CHAPTER 9. THE TOPOLOGY OF METRIC SPACES Informally, we say: The inverse image under f of every open set in Y is an open set in X. If f : X → Y is continuous we occasionally call f a mapping from X to Y . Note that whether or not a particular function f is continuous depends upon the topologies, that is, what the open sets are, of both the domain and range. This is an important property of continuity. Later, when we specialize our study of topological spaces to Metric Spaces, we shall see that our − δ definition of continuity and the topological definition of continuity are the same. A useful way to test continuity of a function is given by the following theorem. Theorem 9.2 f :(X, C) → (Y,C) is continuous on X if and only if for every x ∈ X and every open set V containing f(x) there exists an open set U containing x such that f(U) ⊂ V. Proof: If f : X → Y is continuous on X, then since the inverse of every open set in Y is open in X, for any V ∈C, f −1(V )isopeninX and provides the U. Thus the necessity of the condition is proved. Conversely, suppose that V is open in Y ,andy ∈ V . Then for any x ∈ X such that f(x)=y, by hypothesis there exists an open set U in X, containing x,andsuchthat f(U) ⊂ V. Label the above U by Ux to indicate its dependence upon x,andlet O = Ux. {x:f(x)∈V } Then O is open because it is the union of open sets, and clearly2 O = f −1(V ). Corollary 9.3 Let f : R1 →R1 be any function where R1 =(−∞, ∞) with the usual topology (see Example 4), that is, the open sets are open intervals (a, b) and their arbitrary unions. Then in R1, f is continuous in the − δ sense if and only if f is continuous in the topological sense. 2Provide the details. See Exercise 2. 9.1. ABSTRACT TOPOLOGICAL SPACES 5 1 1 Proof: Suppose f is −δ continuous, and x0 ∈R.LetV be an open set in R , and suppose v = f(x0) ∈ V .SinceV is open, it is the union of open intervals, and hence v ∈ (a, b)for some a, b.Thenlet =min{v − a, b − v}, and note that (v − , v + ) ⊂ (a, b). Since f is assumed continuous in the − δ sense, there exists a δ>0 such that |x − x0| <δimplies |f(x) − f(x0)| <implies x ∈ V. Letting U =(x0 − δ, x0 + δ) in Theorem 9.2 shows that f is continuous in the topological sense. Conversely, suppose f is continuous in the topological sense. Let x be arbitrary, and y = f(x). Let >0. Then since V =(y − , y + ) is an open set in the range, by hypothesis thereexistsanopensetU, containing x, in the domain of f,andsuchthatf(U) ⊂ V. But U open implies there exists an interval (a, b), containing x, which is contained in U.That is, z ∈ (a, b) implies f(z) ∈ V .Nowletδ =min{x − a, b − x} and observe that if |z − x| <δ, then z ∈ (a, b) and hence f(z) ∈ V. But f(z) ∈ V implies |y − f(z)| <.Thus, f is − δ continuous at x. Exercise 1 Compute the four intersections in Example 1. The following two exercise indicate the degree with which continuity is connected to the topology of the spaces involved: Exercise 2 In the proof of Theorem 9.2, why is O = f −1(V )? The following exercises show the intimate connection between topology and continuity. Exercise 3 Let X be an arbitrary set with the discrete topology: C is the set of all subsets of X.Let(Y,C) be an arbitrary topological space. Then f :(X, C) → (Y,C) is continuous, for any f ! Exercise 4 Let Y be an arbitrary set endowed with the trivial topology: C = {∅,Y},and (X, C) an arbitrary topological space. Then f :(X, C) → (Y,C) is continuous, for any f. Exercise 5 Let X =(−∞, ∞)andtheopensetsofX be half-open, half-closed intervals of the form [a, b), for a<b,and their arbitrary unions. Let Y =(−∞, ∞), and the open sets 6 CHAPTER 9. THE TOPOLOGY OF METRIC SPACES of Y be arbitrary. Then prove that f(x)=[x] is continuous(!) [Hint: what does the range of f consist of?] Limit Points and the Derived Set Definition 9.3 Let (X, C) be a topological space, and A ⊂ X. Then x ∈ X is called a limit point of the set A provided every open set O containing x also contains at least one point a ∈ A,witha = x. Definition 9.4 Let (X, C) be a topological space, and A ⊂ X.
Load more

Recommended publications
  • De Rham Cohomology
    De Rham Cohomology 1. Definition of De Rham Cohomology Let X be an open subset of the plane. If we denote by C0(X) the set of smooth (i. e. infinitely differentiable functions) on X and C1(X), the smooth 1-forms on X (i. e. expressions of the form fdx + gdy where f; g 2 C0(X)), we have natural differntiation map d : C0(X) !C1(X) given by @f @f f 7! dx + dy; @x @y usually denoted by df. The kernel for this map (i. e. set of f with df = 0) is called the zeroth De Rham Cohomology of X and denoted by H0(X). It is clear that these are precisely the set of locally constant functions on X and it is a vector space over R, whose dimension is precisley the number of connected components of X. The image of d is called the set of exact forms on X. The set of pdx + qdy 2 C1(X) @p @q such that @y = @x are called closed forms. It is clear that exact forms and closed forms are vector spaces and any exact form is a closed form. The quotient vector space of closed forms modulo exact forms is called the first De Rham Cohomology and denoted by H1(X). A path for this discussion would mean piecewise smooth. That is, if γ : I ! X is a path (a continuous map), there exists a subdivision, 0 = t0 < t1 < ··· < tn = 1 and γ(t) is continuously differentiable in the open intervals (ti; ti+1) for all i.
    [Show full text]
  • 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]
  • Analysis in Metric Spaces Mario Bonk, Luca Capogna, Piotr Hajłasz, Nageswari Shanmugalingam, and Jeremy Tyson
    Analysis in Metric Spaces Mario Bonk, Luca Capogna, Piotr Hajłasz, Nageswari Shanmugalingam, and Jeremy Tyson study of quasiconformal maps on such boundaries moti- The authors of this piece are organizers of the AMS vated Heinonen and Koskela [HK98] to axiomatize several 2020 Mathematics Research Communities summer aspects of Euclidean quasiconformal geometry in the set- conference Analysis in Metric Spaces, one of five ting of metric measure spaces and thereby extend Mostow’s topical research conferences offered this year that are work beyond the sub-Riemannian setting. The ground- focused on collaborative research and professional breaking work [HK98] initiated the modern theory of anal- development for early-career mathematicians. ysis on metric spaces. Additional information can be found at https://www Analysis on metric spaces is nowadays an active and in- .ams.org/programs/research-communities dependent field, bringing together researchers from differ- /2020MRC-MetSpace. Applications are open until ent parts of the mathematical spectrum. It has far-reaching February 15, 2020. applications to areas as diverse as geometric group the- ory, nonlinear PDEs, and even theoretical computer sci- The subject of analysis, more specifically, first-order calcu- ence. As a further sign of recognition, analysis on met- lus, in metric measure spaces provides a unifying frame- ric spaces has been included in the 2010 MSC classifica- work for ideas and questions from many different fields tion as a category (30L: Analysis on metric spaces). In this of mathematics. One of the earliest motivations and ap- short survey, we can discuss only a small fraction of areas plications of this theory arose in Mostow’s work [Mos73], into which analysis on metric spaces has expanded.
    [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]
  • Math 601 Algebraic Topology Hw 4 Selected Solutions Sketch/Hint
    MATH 601 ALGEBRAIC TOPOLOGY HW 4 SELECTED SOLUTIONS SKETCH/HINT QINGYUN ZENG 1. The Seifert-van Kampen theorem 1.1. A refinement of the Seifert-van Kampen theorem. We are going to make a refinement of the theorem so that we don't have to worry about that openness problem. We first start with a definition. Definition 1.1 (Neighbourhood deformation retract). A subset A ⊆ X is a neighbourhood defor- mation retract if there is an open set A ⊂ U ⊂ X such that A is a strong deformation retract of U, i.e. there exists a retraction r : U ! A and r ' IdU relA. This is something that is true most of the time, in sufficiently sane spaces. Example 1.2. If Y is a subcomplex of a cell complex, then Y is a neighbourhood deformation retract. Theorem 1.3. Let X be a space, A; B ⊆ X closed subspaces. Suppose that A, B and A \ B are path connected, and A \ B is a neighbourhood deformation retract of A and B. Then for any x0 2 A \ B. π1(X; x0) = π1(A; x0) ∗ π1(B; x0): π1(A\B;x0) This is just like Seifert-van Kampen theorem, but usually easier to apply, since we no longer have to \fatten up" our A and B to make them open. If you know some sheaf theory, then what Seifert-van Kampen theorem really says is that the fundamental groupoid Π1(X) is a cosheaf on X. Here Π1(X) is a category with object pints in X and morphisms as homotopy classes of path in X, which can be regard as a global version of π1(X).
    [Show full text]
  • Open Sets in Topological Spaces
    International Mathematical Forum, Vol. 14, 2019, no. 1, 41 - 48 HIKARI Ltd, www.m-hikari.com https://doi.org/10.12988/imf.2019.913 ii – Open Sets in Topological Spaces Amir A. Mohammed and Beyda S. Abdullah Department of Mathematics College of Education University of Mosul, Mosul, Iraq Copyright © 2019 Amir A. Mohammed and Beyda S. Abdullah. This article is 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 In this paper, we introduce a new class of open sets in a topological space called 푖푖 − open sets. We study some properties and several characterizations of this class, also we explain the relation of 푖푖 − open sets with many other classes of open sets. Furthermore, we define 푖푤 − closed sets and 푖푖푤 − closed sets and we give some fundamental properties and relations between these classes and other classes such as 푤 − closed and 훼푤 − closed sets. Keywords: 훼 − open set, 푤 − closed set, 푖 − open set 1. Introduction Throughout this paper we introduce and study the concept of 푖푖 − open sets in topological space (푋, 휏). The 푖푖 − open set is defined as follows: A subset 퐴 of a topological space (푋, 휏) is said to be 푖푖 − open if there exist an open set 퐺 in the topology 휏 of X, such that i. 퐺 ≠ ∅,푋 ii. A is contained in the closure of (A∩ 퐺) iii. interior points of A equal G. One of the classes of open sets that produce a topological space is 훼 − open.
    [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]
  • Lecture 13: Basis for a Topology
    Lecture 13: Basis for a Topology 1 Basis for a Topology Lemma 1.1. Let (X; T) be a topological space. Suppose that C is a collection of open sets of X such that for each open set U of X and each x in U, there is an element C 2 C such that x 2 C ⊂ U. Then C is the basis for the topology of X. Proof. In order to show that C is a basis, need to show that C satisfies the two properties of basis. To show the first property, let x be an element of the open set X. Now, since X is open, then, by hypothesis there exists an element C of C such that x 2 C ⊂ X. Thus C satisfies the first property of basis. To show the second property of basis, let x 2 X and C1;C2 be open sets in C such that x 2 C1 and x 2 C2. This implies that C1 \ C2 is also an open set in C and x 2 C1 \ C2. Then, by hypothesis, there exists an open set C3 2 C such that x 2 C3 ⊂ C1 \ C2. Thus, C satisfies the second property of basis too and hence, is indeed a basis for the topology on X. On many occasions it is much easier to show results about a topological space by arguing in terms of its basis. For example, to determine whether one topology is finer than the other, it is easier to compare the two topologies in terms of their bases.
    [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]
  • Advance Topics in Topology - Point-Set
    ADVANCE TOPICS IN TOPOLOGY - POINT-SET NOTES COMPILED BY KATO LA 19 January 2012 Background Intervals: pa; bq “ tx P R | a ă x ă bu ÓÓ , / / calc. notation set theory notation / / \Open" intervals / pa; 8q ./ / / p´8; bq / / / -/ ra; bs; ra; 8q: Closed pa; bs; ra; bq: Half-openzHalf-closed Open Sets: Includes all open intervals and union of open intervals. i.e., p0; 1q Y p3; 4q. Definition: A set A of real numbers is open if @ x P A; D an open interval contain- ing x which is a subset of A. Question: Is Q, the set of all rational numbers, an open set of R? 1 1 1 - No. Consider . No interval of the form ´ "; ` " is a subset of . We can 2 2 2 Q 2 ˆ ˙ ask a similar question in R . 2 Is R open in R ?- No, because any disk around any point in R will have points above and below that point of R. Date: Spring 2012. 1 2 NOTES COMPILED BY KATO LA Definition: A set is called closed if its complement is open. In R, p0; 1q is open and p´8; 0s Y r1; 8q is closed. R is open, thus Ø is closed. r0; 1q is not open or closed. In R, the set t0u is closed: its complement is p´8; 0q Y p0; 8q. In 2 R , is tp0; 0qu closed? - Yes. Chapter 2 - Topological Spaces & Continuous Functions Definition:A topology on a set X is a collection T of subsets of X satisfying: (1) Ø;X P T (2) The union of any number of sets in T is again, in the collection (3) The intersection of any finite number of sets in T , is again in T Alternative Definition: ¨ ¨ ¨ is a collection T of subsets of X such that Ø;X P T and T is closed under arbitrary unions and finite intersections.
    [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]