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Topology DMTH503 Topology DMTH503 Edited by: Dr.Sachin Kaushal TOPOLOGY Edited by Dr. Sachin Kaushal Printed by EXCEL BOOKS PRIVATE LIMITED A-45, Naraina, Phase-I, New Delhi-110028 for Lovely Professional University Phagwara SYLLABUS Topology Objectives: For some time now, topology has been firmly established as one of basic disciplines of pure mathematics. It's ideas and methods have transformed large parts of geometry and analysis almost beyond recognition. In this course we will study not only introduce to new concept and the theorem but also put into old ones like continuous functions. Its influence is evident in almost every other branch of mathematics.In this course we study an axiomatic development of point set topology, connectivity, compactness, separability, metrizability and function spaces. Sr. No. Content 1 Topological Spaces, Basis for Topology, The order Topology, The Product Topology on X * Y, The Subspace Topology. 2 Closed Sets and Limit Points, Continuous Functions, The Product Topology, The Metric Topology, The Quotient Topology. 3 Connected Spaces, Connected Subspaces of Real Line, Components and Local Connectedness, 4 Compact Spaces, Compact Subspaces of Real Line, Limit Point Compactness, Local Compactness 5 The Count ability Axioms, The Separation Axioms, Normal Spaces, Regular Spaces, Completely Regular Spaces 6 The Urysohn Lemma, The Urysohn Metrization Theorem, The Tietze Extension Theorem, The Tychonoff Theorem 7 The Stone-Cech Compactification, Local Finiteness, Paracompactness 8 The Nagata-Smirnov Metrization Theorem, The Smirnov Metrization Theorem 9 Complete Metric Spaces, Compactness in Metric Spaces, Pointwise and Compact Convergence, Ascoli’s Theorem 10 Baire Spaces, Introduction to Dimension Theory CONTENT Unit 1: Topological Spaces 1 Richa Nandra, Lovely Professional University Unit 2: Basis for Topology 29 Richa Nandra, Lovely Professional University Unit 3: The Order Topology 36 Richa Nandra, Lovely Professional University Unit 4: The Product Topology on X × Y 41 Richa Nandra, Lovely Professional University Unit 5: The Subspace Topology 54 Sachin Kaushal, Lovely Professional University Unit 6: Closed Sets and Limit Point 62 Sachin Kaushal, Lovely Professional University Unit 7: Continuous Functions 68 Sachin Kaushal, Lovely Professional University Unit 8: The Product Topology 78 Richa Nandra, Lovely Professional University Unit 9: The Metric Topology 84 Richa Nandra, Lovely Professional University Unit 10: The Quotient Topology 100 Richa Nandra, Lovely Professional University Unit 11: Connected Spaces, Connected Subspaces of Real Line 106 Richa Nandra, Lovely Professional University Unit 12: Components and Local Connectedness 116 Richa Nandra, Lovely Professional University Unit 13: Compact Spaces and Compact Subspace of Real Line 124 Richa Nandra, Lovely Professional University Unit 14: Limit Point Compactness 133 Richa Nandra, Lovely Professional University Unit 15: Local Compactness 138 Sachin Kaushal, Lovely Professional University Unit 16: The Countability Axioms 143 Sachin Kaushal, Lovely Professional University Unit 17: The Separation Axioms 153 Sachin Kaushal, Lovely Professional University Unit 18: Normal Spaces, Regular Spaces and Completely Regular Spaces 164 Sachin Kaushal, Lovely Professional University Unit 19: The Urysohn Lemma 173 Richa Nandra, Lovely Professional University Unit 20: The Urysohn Metrization Theorem 180 Richa Nandra, Lovely Professional University Unit 21: The Tietze Extension Theorem 186 Richa Nandra, Lovely Professional University Unit 22: The Tychonoff Theorem 190 Richa Nandra, Lovely Professional University Unit 23: The Stone-Cech Compactification 195 Sachin Kaushal, Lovely Professional University Unit 24: Local Finiteness and Paracompactness 201 Sachin Kaushal, Lovely Professional University Unit 25: The Nagata-Smirnov Metrization Theorem 209 Richa Nandra, Lovely Professional University Unit 26: The Smirnov Metrization Theorem 214 Richa Nandra, Lovely Professional University Unit 27: Complete Metric Spaces 217 Sachin Kaushal, Lovely Professional University Unit 28: Compactness in Metric Spaces 226 Sachin Kaushal, Lovely Professional University Unit 29: Pointwise and Compact Convergence 238 Richa Nandra, Lovely Professional University Unit 30: Ascoli’s Theorem 243 Richa Nandra, Lovely Professional University Unit 31: Baire Spaces 248 Richa Nandra, Lovely Professional University Unit 32: Introduction to Dimension Theory 253 Richa Nandra, Lovely Professional University Richa Nandra, Lovely Professional University Unit 1: Topological Spaces Unit 1: Topological Spaces Notes CONTENTS Objectives Introduction 1.1 Topology and Different Kinds of Topologies 1.1.1 Topology 1.1.2 Different Kinds of Topologies 1.2 Intersection and Union of Topologies 1.3 Open Set, Closed Set and Closure of a Set 1.3.1 Definition of Open Set and Closed Set 1.3.2 Door Space 1.3.3 Closure of a Set 1.3.4 Properties of Closure of Sets 1.4 Neighborhood 1.5 Dense Set and Boundary Set 1.5.1 Dense Set and No where Dense 1.5.2 Boundary Set 1.6 Separable Space, Limit Point and Derived Set 1.6.1 Separable Space 1.6.2 Limit Point or Accumulation Point or Cluster Point 1.6.3 Derived Set 1.7 Interior and Exterior 1.7.1 Interior Point and Exterior Point 1.7.2 Interior Operator and Exterior Operator 1.7.3 Properties of Interior 1.7.4 Properties of Exterior 1.8 Summary 1.9 Keywords 1.10 Review Questions 1.11 Further Readings Objectives After studying this unit, you will be able to: Describe the concept of topological spaces; LOVELY PROFESSIONAL UNIVERSITY 1 Topology Notes Explain the different kinds of topologies Solve the problems on intersection and union of topologies; Define open set and closed set; Describe the neighborhood of a point and solve related problems; Explain the dense set, separable space and related theorems and problems; Know the concept of limit point and derived set; Define interior and exterior of a set. Introduction Topology is that branch of mathematics which deals with the study of those properties of certain objects that remain invariant under certain kind of transformations as bending or stretching. In simple words, topology is the study of continuity and connectivity. Topology, like other branches of pure mathematics, is an axiomatic subject. In this, we use a set of axioms to prove propositions and theorems. This unit starts with the definition of a topology and moves on to the topics like stronger and weaker topologies, discrete and indiscrete topologies, cofinite topology, intersection and union of topologies, open set and closed set, neighborhood, dense set, etc. 1.1 Topology and Different Kinds of Topologies 1.1.1 Topology Definition 1: Let X be a non-empty set. A collection T of subsets of X is said to be a topology on X if (i) X T, T (ii) the intersection of any two sets in T belongs to T i.e. A T, B T A B T (iii) the union of any (finite or infinite) no. of sets in T belongs to T. i.e. A T UA T where is an arbitrary set. The pair (X, T) is called a Topological space. Example 1: Let X = {p, q, r, s, t, u} and T1 = {X, , {p}, {r, s}, {p, r, s}, {q, r, s, t, u}} Then T1 is a topology on X as it satisfies conditions (i), (ii) and (iii) of definition 1. Example 2: Let X = {a, b, c, d, e} and T2 = {X, , {a}, {c, d}, {a, c, e}, {b, c, d}} Then T2 is not a topology on X as the union of two members of T2 does not belong to T2. {c, d} {a, c, e} = {a, c, d, e} So, T2 does not satisfy condition (iii) of definition 1. 2 LOVELY PROFESSIONAL UNIVERSITY Unit 1: Topological Spaces 1.1.2 Different Kinds of Topologies Notes Stronger and Weaker Topologies Let X be a set and let T1 and T2 be two topologies defined on X. If T1 T2, then T1 is called smaller or weaker topology than T2. If T1 T2, then we also say that T2 is longer or stronger topology than T1. Comparable and Non-comparable Topologies Definitions: The topologies T1 and T2 are said to be comparable if T1 T2 or T2 T1. The topologies T1 and T2 are said to be non-comparable if T1 T2 and T2 T1. Example 3: If X = {s, t} then T1 = {, {s, X}} and T2 = {, {t}, X} are non-comparable as T1 T2 and T2 T1. Discrete and Indiscrete Topology Let X be any non-empty set and T be the collection of all subsets of X. Then T is called the discrete topology on the set X. The topological space (X, T) is called a discrete space. It may be noted that T in above definition satisfy the conditions of definition 1 and so is a topology. Let X be any non-empty set and T = {X, }. Then T is called the indiscrete topology and (X, T) is said to be an indiscrete space. Again, it may be checked that T satisfies the conditions of definition 1 and so is also a topology. Example 4: If X = {a, b, c} and T is a topology on X with {a} T, {b} T, {c} T, prove that T is the discrete topology. Solution: The subsets of X are: X1 = , X2 = {a}, X3 = {b}, X4 = {c}, X5 = {a, b}, X6 = {a, c}, X7 = {b, c}, X8 = {a, b, c} = X In order to prove that T is the discrete topology, we need to prove that each of these subsets belongs to T. As T is a topology, so X and belongs to T. i.e. X1 T, X8 T. Clearly, X2 T, X3 T, X4 T Now X5 = {a, b} = {a} {b} since {a} T, {b} T (Given) and T is a topology and so by definition 1, their union is also in T i.e. X 5 = {a, b} T similarly, X6 = {a, c} = {a} {c} T and X7 = {b, c} = {b} {c} T Hence, T is the discrete topology. LOVELY PROFESSIONAL UNIVERSITY 3 Topology Notes Cofinite Topology Let X be a non-empty set, and let T be a collection of subsets of X whose complements are finite along with , forms a topology on X and is called cofinite topology.
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