DOCSLIB.ORG

Math 550A, Homework 4 Solutions

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Math 550A, Homework 4 Solutions Math 550A, Homework 4 Solutions April 2, 2014 Problem 1. Prove that if {Xα}α∈J is a collection of Hausdorff spaces, then α∈J Xα is Hausdorff in both the product and box topologies. Q Proof. Let {Xα}α∈J be a collection of Hausdorff spaces. Consider (xα)α∈J = (yα)α∈J in α∈J Xα. Then by definition, there exists α0 ∈ J such that xα0 = yα0 in Xα0 . But sinceQ Xα0 is Hausdorff by hypothesis, there exists Uα0 ∋ xα0 and Vα0 ∋ yα0 open in Xα0 such that Uα0 ∩ Vα0 = ∅. But then α∈J (Uα ∩ Vα) such that Uα = Xα and Vα = Xα for all α ∈ J except when α = α0,Q in which case Uα = Uα0 and Vα = Vα0 as defined above. Since Uα0 ∩ Vα0 = ∅, it follows that U ∩ V = (U ∩ V )= ∅ α α α α αY∈J αY∈J αY∈J Observe that U = α∈J Uα and V = α∈J Vα are open in both the box and product topologies. Hence,Q we have shown thatQ given x, y ∈ α∈J Xα such that x = y there exists U ∋ x and V ∋ y open such that U ∩ V = ∅. Q Problem 2. Let R∞ be the subset of Rω consisting of all sequences that are eventually zero, that is, all sequences (x1, x2,...) such that xi = 0 for only finitely many values of i. What is the closure of R∞ in Rω in the box and product topologies? Justify your answer. Proof. In the product topology, R∞ = Rω. Recall that x ∈ A if and only if for all U ∋ x ∞ Rω Rω ∞ open, then U∩A = ∅. Let xn n=1 ∈ and let U be open in such that xn n=1 ∈ U. ∞ R Since U is open in the product topology, U = n=1 Un where Un = for all but finitely many n ∈ N. For the n ∈ N such that Un = QR, Un = (an, bn) where an < bn. Define ∞ R sn n=1 ∈ U as follows: if Un = , sn = 0, otherwise, sn ∈ Un = (an, bn). Observe ∞ R∞ ∞ Rω ∞ that sn =1 ∈ . Hence, for any xn =1 ∈ and any U ∋ xn =1, it follows that n ∞ n n R∞ R∞ R∞ Rω U ∩ = ∅ since sn n=1 ∈ U ∩ . Conclude by definition that = . In the box topology, R∞ = R∞. This means that R∞ is closed, so to show this, Rω R∞ ∞ Rω R∞ it would suffice to show that \ is open. Let xn n=1 ∈ \ . We want to ∞ Rω R∞ show that there exists U ∋ xn n=1 open such that U⊆ \ . Define U as follows: 1 R R ∞ if xn = 0, then Un = , if xn = 0, then Un = \{0} and U = n=1 Un. Observe that all Un are open as we have defined them and thus U is open inQ the box topology. ∞ ∞ Rω R∞ If yn n=1 ∈ U, then yn = 0 if xn = 0 and thus xn n=1 ∈ \ . Conclude that ∞ Rω R∞ Rω R∞ R∞ xn n=1 ∈ U ⊆ \ and thus \ is open and thus is closed and thus R∞= R∞. Problem 3. Given sequences (a1, a2,...) and (b1, b2,...) of real numbers ai > 0 for ω ω all i, define h : R → R by the equation h (x1, x2,...) = (a1x1 + b1, a2x2 + b2,...). Show that if Rω is given the product topology, h is a homeomorphism of Rω with itself. What happens if Rω is given the product topology? Proof. We first show h is a bijection (Note: this is topology invariant). Suppose ∞ ∞ ∞ ∞ N h (cn)n=1 = h (dn)n=1 , then ancn + bn n=1 = andn + bn n=1 and so for any n ∈ , ancn + bn = andn + bn which implies thatcn = dn (since we subtract bn for both sides ∞ ∞ and divide both sides by an since an > 0). Conclude by (cn)n=1 =(dn)n=1 and thus h ∞ Rω ∞ Rω yn−bn is injective. Let (yn)n=1 ∈ and define (xn)n=1 ∈ by xn = an (which is defined ∞ ∞ since an > 0. Then h (xn)n=1 =(yn)n=1 and h is surjective. ∞ Let’s consider the product topology with basis BProd = U : U = R for all n=1 n n Q but finitely many n ∈ N . Let’s consider what h does to basis elements. We show h−1 ∞ B R is continuous by showing that h is an open map. Let U = n=1 Un ∈ Prod. If Un = , then Un =(cn,dn) for some cn <dn and h maps that componentQ to (ancn+bn, andn+bn) which is also open, other wise Un = R and h maps that component to R. Hence the image of a basis element via h is open in the product topology. Conclude that h is open and thus h−1 is continuous. By linearity, similar reasoning can show that h is continuous. Conclude that h is a homeomorphism in the product topology. Similar reasoning shows that h is a homeomorphism in the box topology. 2.
Load more

Recommended publications
  • Generalized Inverse Limits and Topological Entropy
    View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Repository of Faculty of Science, University of Zagreb FACULTY OF SCIENCE DEPARTMENT OF MATHEMATICS Goran Erceg GENERALIZED INVERSE LIMITS AND TOPOLOGICAL ENTROPY DOCTORAL THESIS Zagreb, 2016 PRIRODOSLOVNO - MATEMATICKIˇ FAKULTET MATEMATICKIˇ ODSJEK Goran Erceg GENERALIZIRANI INVERZNI LIMESI I TOPOLOŠKA ENTROPIJA DOKTORSKI RAD Zagreb, 2016. FACULTY OF SCIENCE DEPARTMENT OF MATHEMATICS Goran Erceg GENERALIZED INVERSE LIMITS AND TOPOLOGICAL ENTROPY DOCTORAL THESIS Supervisors: prof. Judy Kennedy prof. dr. sc. Vlasta Matijevic´ Zagreb, 2016 PRIRODOSLOVNO - MATEMATICKIˇ FAKULTET MATEMATICKIˇ ODSJEK Goran Erceg GENERALIZIRANI INVERZNI LIMESI I TOPOLOŠKA ENTROPIJA DOKTORSKI RAD Mentori: prof. Judy Kennedy prof. dr. sc. Vlasta Matijevic´ Zagreb, 2016. Acknowledgements First of all, i want to thank my supervisor professor Judy Kennedy for accept- ing a big responsibility of guiding a transatlantic student. Her enthusiasm and love for mathematics are contagious. I thank professor Vlasta Matijevi´c, not only my supervisor but also my role model as a professor of mathematics. It was privilege to be guided by her for master's and doctoral thesis. I want to thank all my math teachers, from elementary school onwards, who helped that my love for math rises more and more with each year. Special thanks to Jurica Cudina´ who showed me a glimpse of math theory already in the high school. I thank all members of the Topology seminar in Split who always knew to ask right questions at the right moment and to guide me in the right direction. I also thank Iztok Baniˇcand the rest of the Topology seminar in Maribor who welcomed me as their member and showed me the beauty of a teamwork.
    [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]
  • 14. Arbitrary Products
    14. Arbitrary products Contents 1 Motivation 2 2 Background and preliminary definitions2 N 3 The space Rbox 4 N 4 The space Rprod 6 5 The uniform topology on RN, and completeness 12 6 Summary of results about RN 14 7 Arbitrary products 14 8 A final note on completeness, and completions 19 c 2018{ Ivan Khatchatourian 1 14. Arbitrary products 14.2. Background and preliminary definitions 1 Motivation In section 8 of the lecture notes we discussed a way of defining a topology on a finite product of topological spaces. That definition more or less agreed with our intuition. We should expect products of open sets in each coordinate space to be open in the product, for example, and the only issue that arises is that these sets only form a basis rather than a topology. So we simply generate a topology from them and call it the product topology. This product topology \played nicely" with the topologies on each of the factors in a number of ways. In that earlier section, we also saw that every topological property we had studied up to that point other than ccc-ness was finitely productive. Since then we have developed some new topological properties like regularity, normality, and metrizability, and learned that except for normality these are also finitely productive. So, ultimately, most of the properties we have studied are finitely productive. More importantly, the proofs that they are finitely productive have been easy. Look back for example to your proof that separability is finitely productive, and you will see that there was almost nothing to do.
    [Show full text]
  • 3 Limits of Sequences and Filters
    3 Limits of Sequences and Filters The Axiom of Choice is obviously true, the well-ordering theorem is obviously false; and who can tell about Zorn’s Lemma? —Jerry Bona (Schechter, 1996) Introduction. Chapter 2 featured various properties of topological spaces and explored their interactions with a few categorical constructions. In this chapter we’ll again discuss some topological properties, this time with an eye toward more fine-grained ideas. As introduced early in a study of analysis, properties of nice topological spaces X can be detected by sequences of points in X. We’ll be interested in some of these properties and the extent to which sequences suffice to detect them. But take note of the adjective “nice” here. What if X is any topological space, not just a nice one? Unfortunately, sequences are not well suited for characterizing properties in arbitrary spaces. But all is not lost. A sequence can be replaced with a more general construction—a filter—which is much better suited for the task. In this chapter we introduce filters and highlight some of their strengths. Our goal is to spend a little time inside of spaces to discuss ideas that may be familiar from analysis. For this reason, this chapter contains less category theory than others. On the other hand, we’ll see in section 3.3 that filters are a bit like functors and hence like generalizations of points. This perspective thus gives us a coarse-grained approach to investigating fine-grained ideas. We’ll go through some of these basic ideas—closure, limit points, sequences, and more—rather quickly in sections 3.1 and 3.2.
    [Show full text]
  • HOMOTOPY THEORY for BEGINNERS Contents 1. Notation
    HOMOTOPY THEORY FOR BEGINNERS JESPER M. MØLLER Abstract. This note contains comments to Chapter 0 in Allan Hatcher's book [5]. Contents 1. Notation and some standard spaces and constructions1 1.1. Standard topological spaces1 1.2. The quotient topology 2 1.3. The category of topological spaces and continuous maps3 2. Homotopy 4 2.1. Relative homotopy 5 2.2. Retracts and deformation retracts5 3. Constructions on topological spaces6 4. CW-complexes 9 4.1. Topological properties of CW-complexes 11 4.2. Subcomplexes 12 4.3. Products of CW-complexes 12 5. The Homotopy Extension Property 14 5.1. What is the HEP good for? 14 5.2. Are there any pairs of spaces that have the HEP? 16 References 21 1. Notation and some standard spaces and constructions In this section we fix some notation and recollect some standard facts from general topology. 1.1. Standard topological spaces. We will often refer to these standard spaces: • R is the real line and Rn = R × · · · × R is the n-dimensional real vector space • C is the field of complex numbers and Cn = C × · · · × C is the n-dimensional complex vector space • H is the (skew-)field of quaternions and Hn = H × · · · × H is the n-dimensional quaternion vector space • Sn = fx 2 Rn+1 j jxj = 1g is the unit n-sphere in Rn+1 • Dn = fx 2 Rn j jxj ≤ 1g is the unit n-disc in Rn • I = [0; 1] ⊂ R is the unit interval • RP n, CP n, HP n is the topological space of 1-dimensional linear subspaces of Rn+1, Cn+1, Hn+1.
    [Show full text]
  • Math 131: Introduction to Topology 1
    Math 131: Introduction to Topology 1 Professor Denis Auroux Fall, 2019 Contents 9/4/2019 - Introduction, Metric Spaces, Basic Notions3 9/9/2019 - Topological Spaces, Bases9 9/11/2019 - Subspaces, Products, Continuity 15 9/16/2019 - Continuity, Homeomorphisms, Limit Points 21 9/18/2019 - Sequences, Limits, Products 26 9/23/2019 - More Product Topologies, Connectedness 32 9/25/2019 - Connectedness, Path Connectedness 37 9/30/2019 - Compactness 42 10/2/2019 - Compactness, Uncountability, Metric Spaces 45 10/7/2019 - Compactness, Limit Points, Sequences 49 10/9/2019 - Compactifications and Local Compactness 53 10/16/2019 - Countability, Separability, and Normal Spaces 57 10/21/2019 - Urysohn's Lemma and the Metrization Theorem 61 1 Please email Beckham Myers at [email protected] with any corrections, questions, or comments. Any mistakes or errors are mine. 10/23/2019 - Category Theory, Paths, Homotopy 64 10/28/2019 - The Fundamental Group(oid) 70 10/30/2019 - Covering Spaces, Path Lifting 75 11/4/2019 - Fundamental Group of the Circle, Quotients and Gluing 80 11/6/2019 - The Brouwer Fixed Point Theorem 85 11/11/2019 - Antipodes and the Borsuk-Ulam Theorem 88 11/13/2019 - Deformation Retracts and Homotopy Equivalence 91 11/18/2019 - Computing the Fundamental Group 95 11/20/2019 - Equivalence of Covering Spaces and the Universal Cover 99 11/25/2019 - Universal Covering Spaces, Free Groups 104 12/2/2019 - Seifert-Van Kampen Theorem, Final Examples 109 2 9/4/2019 - Introduction, Metric Spaces, Basic Notions The instructor for this course is Professor Denis Auroux. His email is [email protected] and his office is SC539.
    [Show full text]
  • Math 344-1: Introduction to Topology Northwestern University, Lecture Notes
    Math 344-1: Introduction to Topology Northwestern University, Lecture Notes Written by Santiago Can˜ez These are notes which provide a basic summary of each lecture for Math 344-1, the first quarter of “Introduction to Topology”, taught by the author at Northwestern University. The book used as a reference is the 2nd edition of Topology by Munkres. Watch out for typos! Comments and suggestions are welcome. Contents Lecture 1: Topological Spaces 2 Lecture 2: More on Topologies 3 Lecture 3: Bases 4 Lecture 4: Metric Spaces 5 Lecture 5: Product Topology 6 Lecture 6: More on Products 8 Lecture 7: Arbitrary Products, Closed Sets 11 Lecture 8: Hausdorff Spaces 15 Lecture 9: Continuous Functions 16 Lecture 10: More on Continuity 18 Lecture 11: Quotient Spaces 19 Lecture 12: More on Quotients 20 Lecture 13: Connected Spaces 21 Lecture 14: More on Connectedness 22 Lecture 15: Local Connectedness 23 Lecture 16: Compact spaces 24 Lecture 17: More on Compactness 25 Lecture 18: Local Compactness 27 Lecture 19: More on Local Compactness 28 Lecture 20: Countability Axioms 29 Lecture 21: Regular Spaces 30 Lecture 22: Normal spaces 31 Lecture 23: Urysohn’s Lemma 31 Lecture 24: More on Urysohn 32 Lecture 25: Tietze Extension Theorem 32 Lecture 26: Tychonoff’s Theorem 33 Lecture 27: Alexander Subbase Theorem 36 Lecture 1: Topological Spaces Why topology? Topology provides the most general setting in which we can talk about continuity, which is good because continuous functions are amazing things to have available. Topology does this by providing a general setting in which we can talk about the notion of “near” or “close”, and it is this perspective which I hope to make more precise as we go on.
    [Show full text]
  • Topologies of Some New Sequence Spaces, Their Duals, and the Graphical Representations of Neighborhoods ∗ Eberhard Malkowsky A, , Vesna Velickoviˇ C´ B
    View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector Topology and its Applications 158 (2011) 1369–1380 Contents lists available at ScienceDirect Topology and its Applications www.elsevier.com/locate/topol Topologies of some new sequence spaces, their duals, and the graphical representations of neighborhoods ∗ Eberhard Malkowsky a, , Vesna Velickoviˇ c´ b a Department of Mathematics, Faculty of Sciences, Fatih University, 34500 Büyükçekmece, Istanbul, Turkey b Department of Mathematics, Faculty of Sciences and Mathematics, University of Niš, Višegradska 33, 18000 Niš, Serbia article info abstract MSC: Many interesting topologies arise in the theory of FK spaces. Here we present some new 40H05 results on the topologies of certain FK spaces and the determinations of their dual spaces. 54E35 We also apply our software package for visualization and animations in mathematics and its extensions to the representation of neighborhoods in various topologies. Keywords: © 2011 Elsevier B.V. All rights reserved. Topologies and neighborhoods FK spaces Dual spaces Visualization Computer graphics 1. Introduction This paper deals with two subjects in mathematics and computer graphics, namely the study of certain linear topological spaces and the graphical representation of neighborhoods in the considered topologies. Many important topologies arise in the theory of sequence spaces, in particular, in FK and BK spaces and their dual spaces. Here we present some new results on the topology of certain FK spaces and their dual spaces. Our results have an interesting and important application in crystallography, in particular in the growth of crystals. According to Wulff’s principle [1], the shape of a crystal is uniquely determined by its surface energy function.
    [Show full text]
  • Topological Vector Spaces
    An introduction to some aspects of functional analysis, 3: Topological vector spaces Stephen Semmes Rice University Abstract In these notes, we give an overview of some aspects of topological vector spaces, including the use of nets and filters. Contents 1 Basic notions 3 2 Translations and dilations 4 3 Separation conditions 4 4 Bounded sets 6 5 Norms 7 6 Lp Spaces 8 7 Balanced sets 10 8 The absorbing property 11 9 Seminorms 11 10 An example 13 11 Local convexity 13 12 Metrizability 14 13 Continuous linear functionals 16 14 The Hahn–Banach theorem 17 15 Weak topologies 18 1 16 The weak∗ topology 19 17 Weak∗ compactness 21 18 Uniform boundedness 22 19 Totally bounded sets 23 20 Another example 25 21 Variations 27 22 Continuous functions 28 23 Nets 29 24 Cauchy nets 30 25 Completeness 32 26 Filters 34 27 Cauchy filters 35 28 Compactness 37 29 Ultrafilters 38 30 A technical point 40 31 Dual spaces 42 32 Bounded linear mappings 43 33 Bounded linear functionals 44 34 The dual norm 45 35 The second dual 46 36 Bounded sequences 47 37 Continuous extensions 48 38 Sublinear functions 49 39 Hahn–Banach, revisited 50 40 Convex cones 51 2 References 52 1 Basic notions Let V be a vector space over the real or complex numbers, and suppose that V is also equipped with a topological structure. In order for V to be a topological vector space, we ask that the topological and vector spaces structures on V be compatible with each other, in the sense that the vector space operations be continuous mappings.
    [Show full text]
  • Ultrafilters and Tychonoff's Theorem
    Ultrafilters and Tychonoff's Theorem (October, 2015 version) G. Eric Moorhouse, University of Wyoming Tychonoff's Theorem, often cited as one of the cornerstone results in general topol- ogy, states that an arbitrary product of compact topological spaces is compact. In the 1930's, Tychonoff published a proof of this result in the special case of [0; 1]A, also stating that the general case could be proved by similar methods. The first proof in the gen- eral case was published by Cechˇ in 1937. All proofs rely on the Axiom of Choice in one of its equivalent forms (such as Zorn's Lemma or the Well-Ordering Principle). In fact there are several proofs available of the equiv- alence of Tychonoff's Theorem and the Axiom of Choice|from either one we can obtain the other. Here we provide one of the most popular proofs available for Ty- chonoff's Theorem, using ultrafilters. We see this approach as an opportunity to learn also about the larger role of ultra- filters in topology and in the foundations of mathematics, including non-standard analysis. As an appendix, we also include an outline of an alternative proof of Tychonoff's Theorem using a transfinite induction, using the two-factor case X × Y (previously done in class) for the inductive step. This approach is found in the exercises of the Munkres textbook. The proof of existence of nonprincipal ultrafilters was first published by Tarski in 1930; but the concept of ultrafilters is attributed to H. Cartan. 1. Compactness and the Finite Intersection Property Before proceeding, let us recall that a collection of sets S has the finite intersection property if S1 \···\ Sn 6= ? for all S1;:::;Sn 2 S, n > 0.
    [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]