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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.......................... -
Solutions to Exercises in Munkres
April 21, 2006 Munkres §29 Ex. 29.1. Closed intervals [a, b] ∩ Q in Q are not compact for they are not even sequentially compact [Thm 28.2]. It follows that all compact subsets of Q have empty interior (are nowhere dense) so Q can not be locally compact. To see that compact subsets of Q are nowhere dense we may argue as follows: If C ⊂ Q is compact and C has an interior point then there is a whole open interval (a, b) ∩ Q ⊂ C and also [a, b] ∩ Q ⊂ C for C is closed (as a compact subset of a Hausdorff space [Thm 26.3]). The closed subspace [a, b] ∩ Q of C is compact [Thm 26.2]. This contradicts that no closed intervals of Q are compact. Ex. 29.2. Q (a). Assume that the product Xα is locally compact. Projections are continuous and open [Ex 16.4], so Xα is locally compact for all α [Ex 29.3]. Furthermore, there are subspaces U ⊂ C such that U is nonempty and open and C is compact. Since πα(U) = Xα for all but finitely many α, also πα(C) = Xα for all but finitely many α. But C is compact so also πα(C) is compact. Q (b). We have Xα = X1 × X2 where X1 is a finite product of locally compact spaces and X2 is a product of compact spaces. It is clear that finite products of locally compact spaces are locally compact for finite products of open sets are open and all products of compact spaces are compact by Tychonoff. -
Introduction to Algebraic Topology MAST31023 Instructor: Marja Kankaanrinta Lectures: Monday 14:15 - 16:00, Wednesday 14:15 - 16:00 Exercises: Tuesday 14:15 - 16:00
Introduction to Algebraic Topology MAST31023 Instructor: Marja Kankaanrinta Lectures: Monday 14:15 - 16:00, Wednesday 14:15 - 16:00 Exercises: Tuesday 14:15 - 16:00 August 12, 2019 1 2 Contents 0. Introduction 3 1. Categories and Functors 3 2. Homotopy 7 3. Convexity, contractibility and cones 9 4. Paths and path components 14 5. Simplexes and affine spaces 16 6. On retracts, deformation retracts and strong deformation retracts 23 7. The fundamental groupoid 25 8. The functor π1 29 9. The fundamental group of a circle 33 10. Seifert - van Kampen theorem 38 11. Topological groups and H-spaces 41 12. Eilenberg - Steenrod axioms 43 13. Singular homology theory 44 14. Dimension axiom and examples 49 15. Chain complexes 52 16. Chain homotopy 59 17. Relative homology groups 61 18. Homotopy invariance of homology 67 19. Reduced homology 74 20. Excision and Mayer-Vietoris sequences 79 21. Applications of excision and Mayer - Vietoris sequences 83 22. The proof of excision 86 23. Homology of a wedge sum 97 24. Jordan separation theorem and invariance of domain 98 25. Appendix: Free abelian groups 105 26. English-Finnish dictionary 108 References 110 3 0. Introduction These notes cover a one-semester basic course in algebraic topology. The course begins by introducing some fundamental notions as categories, functors, homotopy, contractibility, paths, path components and simplexes. After that we will study the fundamental group; the Fundamental Theorem of Algebra will be proved as an application. This will take roughly the first half of the semester. During the second half of the semester we will study singular homology. -
General Topology
General Topology Andrew Kobin Fall 2013 | Spring 2014 Contents Contents Contents 0 Introduction 1 1 Topological Spaces 3 1.1 Topology . .3 1.2 Basis . .5 1.3 The Continuum Hypothesis . .8 1.4 Closed Sets . 10 1.5 The Separation Axioms . 11 1.6 Interior and Closure . 12 1.7 Limit Points . 15 1.8 Sequences . 16 1.9 Boundaries . 18 1.10 Applications to GIS . 20 2 Creating New Topological Spaces 22 2.1 The Subspace Topology . 22 2.2 The Product Topology . 25 2.3 The Quotient Topology . 27 2.4 Configuration Spaces . 31 3 Topological Equivalence 34 3.1 Continuity . 34 3.2 Homeomorphisms . 38 4 Metric Spaces 43 4.1 Metric Spaces . 43 4.2 Error-Checking Codes . 45 4.3 Properties of Metric Spaces . 47 4.4 Metrizability . 50 5 Connectedness 51 5.1 Connected Sets . 51 5.2 Applications of Connectedness . 56 5.3 Path Connectedness . 59 6 Compactness 63 6.1 Compact Sets . 63 6.2 Results in Analysis . 66 7 Manifolds 69 7.1 Topological Manifolds . 69 7.2 Classification of Surfaces . 70 7.3 Euler Characteristic and Proof of the Classification Theorem . 74 i Contents Contents 8 Homotopy Theory 80 8.1 Homotopy . 80 8.2 The Fundamental Group . 81 8.3 The Fundamental Group of the Circle . 88 8.4 The Seifert-van Kampen Theorem . 92 8.5 The Fundamental Group and Knots . 95 8.6 Covering Spaces . 97 9 Surfaces Revisited 101 9.1 Surfaces With Boundary . 101 9.2 Euler Characteristic Revisited . 104 9.3 Constructing Surfaces and Manifolds of Higher Dimension . -
MAT327H1: Introduction to Topology
MAT327H1: Introduction to Topology Topological Spaces and Continuous Functions TOPOLOGICAL SPACES Definition: Topology A topology on a set X is a collection T of subsets of X , with the following properties: 1. ∅ , X ∈T . ∈ ∈ ∪u ∈T 2. If u T , A , then . ∈A n ∈ = 3. If ui T ,i 1, , n , then ∩ui ∈T . i=1 The elements of T are called open sets. Examples 1. T={∅ , X } is an indiscrete topology. 2. T=2X =set of all subsets of X is a discrete topology. Definition: Finer, Coarser T 1 is finer than T 2 if T 2 ⊂T 1 . T 2 is coarser than T 1 . BASIS FOR A TOPOLOGY Definition: Basis A collection of subsets B of X is called a bais for a topology if: 1. The union of the elements of B is X . 2. If x∈B1 ∩B2 , B1, B2∈B , then there exists a B3 of B such that x∈B3⊂B1∩B2 . Examples 1 1 1. B is the set of open intervals a ,b in ℝ with ab . For each x∈ℝ , x∈ x− , x . 2 2 2. B is the set of all open intervals a ,b in ℝ where ab and a and b are rational numbers. 3. Let T be the collection of subsets of ℝ which are either empty or are the complements of finite sets. Note that c A∪B = Ac∩Bc and A∩Bc= Ac∪Bc . This topology does not have a countable basis. Claim A basis B generates a topology T whose elements are all possible unions of elements of B . -
Assam University Msc Mathematics Syllabus.Pdf
ASSAM UNIVERSITY: SILCHAR Department of Mathematics School of Physical Sciences (CBCS (AUS)) Structure of PG(M.Sc.) Syllabus in Mathematics (To be implemented from Academic Session 2010-2011) SEMESTER-II Paper No. Paper Name Marks L P T C Internal External Total M P. M. M P.M. M201 Topology 25 10 75 30 100 4 1 5 M202 Classical Mechanics 25 10 75 30 100 4 1 5 M203 Discrete Mathematics 25 10 75 30 100 4 1 5 M204 Real Analysis-II 25 10 75 30 100 4 1 5 M205 Partial Differential 25 10 45 18 70+30 4 1 5 (Theory) Equations ( Theory ) M205 Practical-II (External) 30(5+5+20) 12 100 3 (Practical-II) MATLAB/Mathematica/ (Practical MAPLE etc. Notebook + viva-voce + Experiment) Total 125 50 375 150 500 20 03 5 25 SEMESTER-III Paper No. Paper Name Marks L P T C Internal External Total M P. M. M P.M. M301 Linear Algebra 25 10 75 30 100 4 1 5 M302 Fluid Mechanics 25 10 75 30 100 4 1 5 M303 Complex Analysis 25 10 75 30 100 4 1 5 M304 Operations Research 25 10 75 30 100 4 1 5 M305 Mathematical Modelling 25 10 45 18 70+30 4 1 5 (Theory) M305 Practical-III (External) 30(5+5+20) 12 100 3 (Practical- MATLAB/Mathematica/ (Practical III) MAPLE etc. Notebook + viva-voce + Experiment) Total 125 50 375 150 500 20 03 5 25 SEMESTER-IV Paper No. Paper Name Marks L P T C Internal External Total M P. -
Notes on Point-Set Topology
Title. Tyrone Cutler April 29, 2020 Abstract Contents 1 Proper Maps 1 2 Local Compactness 3 3 The Compact-Open Topology 6 3.1 Further Properties of the Compact-Open Topology . 12 1 Proper Maps Proper maps make several appearances in our work. Since they can be very powerful when recognised we include a few results as background. A general reference for this section is Bourbaki [1], x1, Chapter 10. Definition 1 A continuous map f : X ! Y is said to be proper if it is closed, and if for −1 each y 2 Y the inverse image f (y) ⊆ X is compact. Lemma 1.1 If f : X ! Y is proper and K ⊆ Y is compact, then f −1(K) ⊆ X is compact. −1 Proof It will suffice to show that if fCi ⊆ f (K)gi2I is a family of closed subsets with T I Ci = ;, then there is finite subfamily whose intersection is already empty. To this end we consider the family ff(Ci) ⊆ KgI . Since f is proper this is a collection of closed subsets of K ⊆ Y whose total intersection is empty. Since K is compact, there must be finitely many of these sets, say, f(C1); : : : ; f(Cn) such that f(C1) \···\ f(Cn) = ;. But this tells us that f(C1 \···\ Cn) = ;, and the only way this can happen is if C1 \···\ Cn = ;, which was what we needed to show. Proposition 1.2 A map f : X ! Y is proper if and only if for each space Z, the map f × idZ : X × Z ! Y × Z is closed. -
Class Notes for Math 871: General Topology, Instructor Jamie Radcliffe
University of Nebraska - Lincoln DigitalCommons@University of Nebraska - Lincoln Math Department: Class Notes and Learning Materials Mathematics, Department of 2010 Class Notes for Math 871: General Topology, Instructor Jamie Radcliffe Laura Lynch University of Nebraska-Lincoln, [email protected] Follow this and additional works at: https://digitalcommons.unl.edu/mathclass Part of the Science and Mathematics Education Commons Lynch, Laura, "Class Notes for Math 871: General Topology, Instructor Jamie Radcliffe" (2010). Math Department: Class Notes and Learning Materials. 10. https://digitalcommons.unl.edu/mathclass/10 This Article is brought to you for free and open access by the Mathematics, Department of at DigitalCommons@University of Nebraska - Lincoln. It has been accepted for inclusion in Math Department: Class Notes and Learning Materials by an authorized administrator of DigitalCommons@University of Nebraska - Lincoln. Class Notes for Math 871: General Topology, Instructor Jamie Radcliffe Topics include: Topological space and continuous functions (bases, the product topology, the box topology, the subspace topology, the quotient topology, the metric topology), connectedness (path connected, locally connected), compactness, completeness, countability, filters, and the fundamental group. Prepared by Laura Lynch, University of Nebraska-Lincoln August 2010 1 Topological Spaces and Continuous Functions Topology is the axiomatic study of continuity. We want to study the continuity of functions to and from the spaces C; Rn;C[0; 1] = ff : [0; 1] ! Rjf is continuousg; f0; 1gN; the collection of all in¯nite sequences of 0s and 1s, and H = 1 P 2 f(xi)1 : xi 2 R; xi < 1g; a Hilbert space. De¯nition 1.1. A subset A of R2 is open if for all x 2 A there exists ² > 0 such that jy ¡ xj < ² implies y 2 A: 2 In particular, the open balls B²(x) := fy 2 R : jy ¡ xj < ²g are open. -
MATHEMATICS I Semester Course No. Course Title Lec Hr Tut Hr SS Hr Lab Hr DS Hr AL TC Hr Grading System Credits (AL/3) MTH
MATHEMATICS I Semester Lec Tut SS Lab DS AL TC Grading Credits Course No. Course Title Hr Hr Hr Hr Hr Hr System (AL/3) MTH 101 Calculus of One Variable 3 1 4.5 0 0 8.5 4 O to F 3 II Semester Lec Tut SS Lab DS AL TC Grading Credits Course No. Course Title Hr Hr Hr Hr Hr Hr System (AL/3) MTH 102 Linear Algebra 3 1 4.5 0 0 8.5 4 O to F 3 III Semester Lec Tut SS Lab DS AL TC Grading Credits Course No. Course Title Hr Hr Hr Hr Hr Hr System (AL/3) Mathematics MTH201 Multivariable Calculus 3 1 4.5 0 0 8.5 4 O to F 3 DC MTH203 Introduction to Groups and Symmetry 3 1 4.5 0 0 8.5 4 O to F 3 IV Semester Course Lec Tut SS Lab DS AL TC Grading Credits Course Title No. Hr Hr Hr Hr Hr Hr System (AL/3) Mathematics MTH202 Probability and Statistics 3 1 4.5 0 0 8.5 4 O to F 3 DC MTH204 Complex Variables 3 1 4.5 0 0 8.5 4 O to F 3 DC: Departmental Compulsory Course V Semester Course No. Course Title Lec Tut SS Hr Lab DS AL TC Grading Credits Hr Hr Hr Hr Hr System MTH 301 Group Theory 3 0 7.5 0 0 10.5 3 O to F 4 MTH 303 Real Analysis I 3 0 7.5 0 0 10.5 3 O to F 4 MTH 305 Elementary Number Theory 3 0 7.5 0 0 10.5 3 O to F 4 MTH *** Departmental Elective I 3 0 7.5 0 0 10.5 3 O to F 4 *** *** Open Elective I 3 0 4.5/7.5 0 0 7.5/10.5 3 O to F 3/4 Total Credits 15 0 34.5/37.5 0 0 49.5/52.5 15 19/20 VI Semester Course No. -
23. Mon, Oct. 21 Like Connectedness, Compactness Is Preserved by Continuous Functions
23. Mon, Oct. 21 Like connectedness, compactness is preserved by continuous functions. Proposition 23.1. Let f : X Y be continuous, and assume that X is compact. Then f(X) is compact. ! Proof. Let be an open cover of f(X). Then = f 1(V ) V is an open cover of X. Let V U { − | 2V} U1,...,Uk be a finite subcover. It follows that the corresponding V1,...,Vk is a finite subcover of{ . } { } V ⌅ Example 23.2. Recall that we have the quotient map exp : [0, 1] S1. It follows that S1 is compact. ! Theorem 23.3 (Extreme Value Theorem). Let f :[a, b] R be continuous. Then f attains a maximum and a minimum. ! Proof. Since f is continuous and [a, b] is both connected and compact, the same must be true of its image. But the compact, connected subsets are precisely the closed intervals. ⌅ The following result is also quite useful. Proposition 23.4. Let X be Hausdor↵and let A X be a compact subset. Then A is closed in X. ✓ Proof. Pick any point x X A (if we can’t, then A = X and we are done). For each a A,we have disjoint neighborhoods2 a\ U and x V .SincetheU cover A, we only need finitely2 many, 2 a 2 a a say Ua1 ,...,Uak to cover A. But then the intersection V = V V a1 \···\ ak of the corresponding Va’s is disjoint from the union of the Ua’s and therefore also from A.Since V is a finite intersection of open sets, it is open and thus gives a neighborhood of x in X A.It \ follows that A is closed. -
The Axiom of Choice
Virginia Commonwealth University VCU Scholars Compass Theses and Dissertations Graduate School 2010 The Axiom of Choice Cristian Allen Virginia Commonwealth University Follow this and additional works at: https://scholarscompass.vcu.edu/etd Part of the Physical Sciences and Mathematics Commons © The Author Downloaded from https://scholarscompass.vcu.edu/etd/2145 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]. College of Humanities and Sciences Virginia Commonwealth University This is to certify that the thesis prepared by Cristian Gerardo Allen titled “The Axiom of Choice” has been approved by his or her committee as satisfactory completion of the thesis requirement for the degree of Master of Science. Dr. Andrew Lewis, College of Humanities and Sciences Dr. Kevin Beanland, College of Humanities and Sciences Dr. David Ameen, College of Humanities and Sciences Dr. John Berglund, Graduate Chair, Mathematics and Applied Mathematics Dr. Robert Holsworth, Dean, College of Humanities and Sciences Dr. F. Douglas Boudinot, Graduate Dean Date © Cristian Gerardo Allen 2010 All Rights Reserved The Axiom of Choice A thesis submitted in partial fulfillment of the requirements for the degree of Master of Science at Virginia Commonwealth University. by Cristian Gerardo Allen Master of Science Director: Dr. Andrew Lewis, Associate Professor, Department Chair Department of Mathematics and Applied Mathematics Virginia Commonwealth University Richmond, Virginia May 2010 ii Acknowledgment I am grateful that Virginia Commonwealth University has given me such a strong education in mathematics. -
[Math.FA] 4 Apr 2017 Open Quotients of Trivial Vector Bundles
Open quotients of trivial vector bundles∗ Pedro Resende and Joao˜ Paulo Santos Abstract Given an arbitrary topological complex vector space A, a quo- tient vector bundle for A is a quotient of a trivial vector bundle π2 : A × X → X by a fiberwise linear continuous open surjection. We show that this notion subsumes that of a Banach bundle over a locally compact Hausdorff space X. Hyperspaces consisting of lin- ear subspaces of A, topologized with natural topologies that include the lower Vietoris topology and the Fell topology, provide classifying spaces for various classes of quotient vector bundles, in a way that generalizes the classification of locally trivial vector bundles by Grass- mannians. If A is normed, a finer hyperspace topology is introduced that classifies bundles with continuous norm, including Banach bun- dles, and such that bundles of constant finite rank must be locally trivial. Keywords: Vector bundles, Banach bundles, Grassmannians, lower Vietoris topology, Fell topology. 2010 Mathematics Subject Classification: 46A99, 46M20, 54B20, 55R65 Contents 1 Introduction 2 arXiv:1510.06329v3 [math.FA] 4 Apr 2017 2 Preliminaries 4 3 Quotient vector bundles 7 4 Banach bundles 14 ∗Work funded by FCT/Portugal through projects EXCL/MAT-GEO/0222/2012 and PEst-OE/EEI/LA0009/2013, and by COST (European Cooperation in Science and Tech- nology) through COST Action MP1405 QSPACE. 1 5 Classifying spaces 17 6 Finite rank and local triviality 24 7 Grassmannians 32 A Appendix 37 1 Introduction ∗ The reduced C*-algebra Cred(G) of a locally compact Hausdorff ´etale groupoid G (see [10,11]), whose elements can be regarded as complex valued functions ∗ on G, can be “twisted” by considering instead the reduced C*-algebra Cred(π) of a Fell line bundle π : E → G, where now the elements of the algebra are identified with sections of the bundle; see [5, 12].