دانلود کتاب Basic Topology 1: Metric Spaces and General Topology

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Preface
A Note on Basic Topology—Volumes 1–3
Basic Topology—Volume 1: Metric Spaces and General Topology
Basic Topology—Volume 2: Topological Groups, Topology of Manifolds and Lie Groups
Basic Topology—Volume 3: Algebraic Topology and Topology of Fiber Bundles
Contents
About the Authors
1 Prerequisites: Sets, Algebraic Systems and Classical Analysis
1.1 Sets and Set Operations
1.1.1 Indexed Family of Sets
1.1.2 Set Operations: Union, Intersection and Complement
1.2 Basic Properties of Real Numbers
1.2.1 Least Upper Bound or Completeness Property of R
1.2.2 Archimedean Property of R
1.2.3 Denseness Property of Q in R
1.2.4 Complex Numbers: Ordered Pairs of Real Numbers
1.3 Binary Relations on Sets
1.3.1 Equivalence Relation
1.3.2 Order Relation
1.3.3 Partial Order Relation and Zorn\'s Lemma
1.4 Functions or Mappings
1.4.1 Geometrical Examples of Functions
1.4.2 Geometrical Examples of Bijections
1.4.3 Restriction, Extension and Composition of Functions
1.4.4 Cartesian Product of Any Collection of Sets
1.4.5 Choice Function and Axiom of Choice
1.5 Cantor Set
1.5.1 Construction of the Cantor Set
1.5.2 Geometrical Method of Construction of the Cantor Set
1.6 Countability and Cardinality of Sets
1.6.1 Countability of Sets
1.6.2 Cardinality of Sets
1.6.3 Order Relations on Cardinal Numbers
1.6.4 Sum, Product and Powers of Cardinal Numbers
1.7 Well-Ordered Sets and Ordinal Numbers
1.8 Rings and Ideals
1.9 Vector Space and Linear Transformations
1.9.1 Basis for a Vector Space
1.9.2 Linear Transformations
1.9.3 Dual Space of a Vector Space
1.9.4 mathcalC([0, 1]) as a Vector Space
1.10 Euclidean Spaces and Related Spaces with Standard Notations
1.11 Exercises
References
2 Metric Spaces and Normed Linear Spaces
2.1 Results of Analysis Leading to the Concept of Metric Spaces
2.1.1 Open Sets in R
2.1.2 Closed Sets in R
2.1.3 Two Classical Theorems: Bolzano–Weierstrass Theorem and Heine–Borel Theorem
2.1.4 Continuity of Functions on R
2.2 Sequence of Real Numbers and Cauchy Sequence
2.2.1 Sequence of Real Numbers
2.2.2 Cauchy Sequence
2.3 Concept of Distance in Euclidean Spaces Rn
2.3.1 Open Sets of Euclidean Plane R2
2.3.2 Distance Function in Rn
2.3.3 Continuity of Functions f: RnrightarrowRm
2.4 Metric Spaces: Introductory Concepts
2.5 Examples of Metrics Arising from Mathematical Analysis
2.5.1 Norm Function
2.5.2 Euclidean Metric, lp -Metric and l infty-Metric on Rn
2.5.3 lp -Metric and l infty-Metric on mathcalC([0,1])
2.5.4 lp -Metric and l infty-Metrics on Sequences of Real Numbers
2.5.5 p-Adic Metric on Q
2.6 Open Balls and Open Sets in Metric Spaces
2.7 Neighborhoods in Metric Spaces
2.7.1 Neighborhoods in R
2.7.2 Neighborhoods and Open Sets in Metric Spaces
2.8 Limit Points, Closed and Dense Sets in Metric Spaces
2.9 Diameter of Sets and Continuity of Distance Functions on Metric Spaces
2.9.1 Distance of a Point from a Set in Metric Spaces
2.9.2 Distance Between Two Sets in Metric Spaces
2.9.3 Diameter of a Set
2.9.4 Continuity of Distance Function on a Metric Space
2.10 Sequences, Convergence of Sequences and Cauchy Sequences in Metric Spaces
2.10.1 Convergence of Sequences in Metric Spaces
2.10.2 Cauchy Sequences in Metric Spaces
2.11 Complete Metric Spaces and Cantor\'s Intersection Theorem
2.11.1 Complete Metric Spaces
2.11.2 Completion of a Noncomplete Metric Space
2.11.3 Cantor\'s Intersection Theorem for Metric Spaces
2.12 Continuity and Uniform Continuity in Metric Spaces
2.12.1 Continuity of Functions and Convergence of Sequences in Metric Spaces
2.12.2 Uniform Continuity and Lipschitz Functions in Metric Spaces
2.13 Homeomorphism and Isometry in Metric Spaces
2.13.1 Homeomorphisms in Metric Spaces
2.13.2 Isometry in Metric Spaces
2.13.3 Equivalent Metrics
2.14 Normed Linear Spaces: Banach Spaces, Hilbert Spaces and Hahn–Banach Theorem
2.14.1 Pseudo-normed Linear Spaces
2.14.2 Banach Spaces and Hahn–Banach Theorem
2.14.3 lp -Space
2.14.4 Hilbert Spaces and Examples
2.15 Continuity of Functions on Normed Linear Spaces
2.15.1 Continuous Linear Transformations
2.15.2 Continuous Linear Functionals
2.16 Applications
2.16.1 Banach Contraction Principle
2.16.2 Further Application to Analysis: Picards\'s Theorem
2.16.3 Urysohn Function and Urysohn Lemma for Metric Spaces
2.16.4 Geometrical Applications
2.16.5 Separable Metric Spaces
2.17 Compact Subsets of Metric Spaces
2.17.1 Compactness of Metric Spaces is a Topological Property
2.17.2 Lebesgue Lemma and Lebesgue Number
2.17.3 A Characterization of Totally Bounded Complete Metric Spaces
2.17.4 Connectedness in Metric Spaces and Connected Subsets of R
2.17.5 Connectedness of Metric Spaces is a Topological Property
2.17.6 Other Applications
2.18 Exercises
References
3 Topological Spaces and Continuous Maps
3.1 Topological Spaces: Introductory Concepts
3.1.1 Natural Topology on R, R2 and Rn
3.1.2 Construction of Topologies on Some Finite Sets
3.1.3 Comparison of Topologies
3.1.4 Neighborhoods and Limit Points
3.1.5 Closed Sets
3.1.6 Closed and Open (Clopen) Sets
3.1.7 Closure of a Set
3.1.8 Interior of a Set
3.1.9 Exterior of a Set
3.1.10 Boundary of a Set
3.1.11 Interrelations Among Closure, Interior, Exterior and Boundary Operators
3.1.12 Subspace Topology
3.1.13 Dense and Nowhere Dense Sets
3.2 Open Base and Subbase for a Topology
3.2.1 Open Base
3.2.2 Local Base at a Point in a Topological Space
3.2.3 Subbase for a Topology
3.3 Euclidean Topology
3.3.1 Euclidean Topology on R
3.3.2 Euclidean Topology on R 2
3.3.3 Euclidean Topology on Rn
3.3.4 Special Examples of Open and Closed Sets in Rn
3.4 Topology on Linearly Ordered Sets
3.4.1 Order Topologies on R
3.4.2 Order Topologies on Q and Z
3.4.3 Ordinal Space
3.5 Lattice of Topologies
3.6 Continuous Maps
3.6.1 Problems Leading to Continuous Functions
3.6.2 Continuous Functions: Introductory Concepts
3.6.3 Neighborhoods and Continuity at a Point
3.6.4 Pasting or Gluing Lemma
3.6.5 Path in a Topological Space
3.6.6 Open and Closed Maps
3.6.7 Lebesgue Sets of a Continuous Function
3.7 Homeomorphism, Topological Embedding, Topological …
3.7.1 Problems Leading to Homeomorphism
3.7.2 Homeomorphism
3.7.3 Embedding of Topological Spaces
3.7.4 Topological Property
3.7.5 Topological Invariant with a Historical Note
3.8 Metric Topology and Metrizability of Topological Spaces
3.8.1 Metric Topology
3.8.2 Equivalent Metrics from Viewpoint Topology
3.8.3 Metrizable Spaces
3.8.4 Metrizability Is a Topological Property
3.8.5 Topologically Complete Metric Spaces
3.9 Topology Generated by a Family of Functions
3.10 Kuratowski Closure Topology, Sierpinski Topology and Niemytzki\'s Disk Topology
3.10.1 Kuratowski Closure Topology
3.10.2 Sierpinski Space
3.10.3 Niemytzki Topology
3.11 Two Countability and Separability Axioms
3.11.1 Countability Axioms: First and Second Countable Spaces
3.11.2 Separability Axioms: Separable Spaces
3.12 Sum and Product of Topological Spaces
3.12.1 Sum of Topological Spaces
3.12.2 Product Space of a Finite Family of Topological Spaces
3.12.3 Product Space of an Arbitrary Family of Topological Spaces and Tychonöff Topology
3.13 Topological Product of Metrizable Spaces
3.14 Continuous Maps into Product Spaces
3.15 Weak Topology and Construction of Sinfty, RPinfty and CPinfty
3.15.1 Weak Topology (Union Topology)
3.15.2 Construction of Sinfty, RPinfty and CPinfty with Weak Topology
3.16 Quotient Spaces: Construction of Geometrical Objects …
3.16.1 Quotient Topology and Quotient Spaces
3.16.2 Quotient or Identification Maps
3.16.3 Construction of Sn and R Pn by Identification Method
3.16.4 Constructions of Spheres and Cones by Collapsing Method
3.16.5 Construction of New Spaces by Gluing Method
3.16.6 Attaching Map: Construction of Mapping Cylinder and Mapping Cone
3.16.7 Construction of Cylinder, Möbius Band, Torus and Klein Bottle
3.17 Zariski Topology, Scheme and Zariski Space
3.17.1 Zariski Topology on an Affine Space
3.17.2 Zariski Topology on the Spectrum of a Ring and Scheme
3.17.3 Zariski Space Defined by Descending Chain of Closed Sets
3.18 Topological Applications
3.18.1 Topological Applications in Matrix Algebra
3.18.2 Uniform Convergence of Sequence of Functions to Metric Space
3.18.3 Solution of Homeomorphism Problems in R by Cardinality
3.18.4 Cantor Space
3.18.5 Application of Pasting Lemma for Functions from Product Spaces
3.19 Historical Note: Beginning of Topology Through the Work of Euler
3.19.1 Seven Bridge Problem of Königsberg
3.19.2 Euler Characteristic of a Polyhedron
3.20 Exercises
References
4 Separation Axioms
4.1 Separation by Open Sets and Ti-Spaces
4.1.1 Separation by Open Sets
4.1.2 Separation Axioms and Ti-Spaces
4.1.3 Characterization of T1-Spaces
4.2 Hausdorff Spaces
4.2.1 Basic Properties of Hausdorff Spaces
4.2.2 Separation Property of Left-Hand (Right-Hand) Topology on R
4.2.3 Separation Property of Lower-Limit (Upper-Limit) Topology on R
4.2.4 First Countable Hausdorff Spaces
4.3 Structures of Normal and Completely Normal Spaces
4.3.1 Normal Spaces and Normality Criterion of Urysohn
4.3.2 Completely Normal Spaces
4.4 Structures of Regular and Completely Regular Spaces
4.4.1 Regular Spaces
4.4.2 Independence of Regularity and Hausdorf Properties
4.4.3 Completely Regular Spaces
4.5 Homeomorphisms of Ti-Spaces and Topological Property
4.6 Applications
4.6.1 Minor Urysohn Lemma for Normal Spaces
4.6.2 Link Between Hausdorff Property and Continuity of Real Functions
4.6.3 Hausdorff Property of Rn, Sn , and Hilbert Cube
4.6.4 Retraction of a Hausdorff Space
4.6.5 Locally Euclidean Spaces
4.7 Exercises
References
5 Compactness and Connectedness
5.1 Different Types of Compactness and Compact Subsets of Rn
5.1.1 Motivation of Six Different Types of Compactness
5.1.2 Compactness of Subsets of R and Rn
5.1.3 Compact Sets in Arbitrary Topological Spaces
5.1.4 Subspaces of Compact Spaces
5.2 Compactness in Metric Spaces
5.2.1 Compact Sets in Metric Spaces
5.2.2 Lebesgue Lemma and Lebesgue Number
5.3 Bolzano–Weierstrass (B–W) Compactness
5.4 Basic Link between Compactness and Hausdorff Properties of Topological Spaces
5.5 Sequentially Compact Spaces
5.6 Locally Compact and Compactly Generated Hausdorff Spaces
5.6.1 Locally Compact Spaces
5.6.2 Locally Compact Hausdorff Space
5.6.3 Category of Compactly Generated Hausdorff Spaces
5.7 Baire Space
5.8 Compactness Is a Topological Property
5.9 Characterization of Compactness by Finite Intersection Property with Motivation
5.10 Paracompact Spaces
5.11 Alexander\'s Subbase Theorem and Tychonoff Product Theorem
5.12 Net and Convergence
5.12.1 Net: Introductory Concepts
5.12.2 Convergence
5.13 Compactification Problems: Stone-Čech Compactification …
5.13.1 Motivation of Compactification
5.13.2 Stone-Čech Compactification
5.13.3 Alexandroff One-point Compactification
5.14 Haar–Konig Theorem: Characterization of Compactness in Linearly Ordered Spaces
5.15 Compact Subsets of Metrizable Spaces
5.16 Countably Compactness and Its Characterization
5.17 Connectedness
5.17.1 Three Different Types of Connectedness
5.17.2 Connectedness: Introductory Concepts
5.18 Connectedness Property of R and Its Subspaces
5.19 Disconnected and Totally Disconnected Spaces
5.20 Components of a Topological Space
5.21 Local Connectedness and Its Characterization
5.22 Path Connectedness, Path Component and Locally Path Connectedness
5.23 Space-Filling Curve Theorem
5.24 Function Spaces
5.24.1 Compact Open Topology
5.24.2 Uniform Convergence Topology
5.24.3 Point Open Topology
5.24.4 Pointwise Convergence Topology
5.24.5 Relations on Different Topologies on mathcalF (X,Y)
5.24.6 Ascoli\'s Theorem on Function Spaces
5.25 Applications
5.25.1 Geometric Applications
5.25.2 Matrix Algebra from Viewpoint of Connectedness and Compactness
5.25.3 Topological Study of Algebraic Groups
5.25.4 Applications to Homeomorphism Problems
5.25.5 Alternative Proof of Heine–Borel Theorem and Compactness of Sn
5.26 Application in Measure Theory
5.27 Further Applications
5.27.1 Some Applications of Real-valued Continuous Functions
5.27.2 Brouwer Fixed Point Theorem for Dimension 1
5.28 Exercises
References
6 Real-Valued Continuous Functions
6.1 Real-Valued Continuous Functions: Introductory Concepts
6.1.1 Continuity of Real-Valued Functions
6.1.2 Uniform Convergence of Real-Valued Functions
6.1.3 mathcalC (X,R) and mathcalB (X,R)
6.2 Urysohn Lemma: Separation of Disjoint Subsets of Topological Spaces
6.2.1 Existence of Real-Valued Continuous Functions
6.2.2 Functionally Separable Sets
6.2.3 Urysohn Lemma and Characterization of Normal Spaces
6.3 More on Completely Regular and Tychonoff Spaces
6.3.1 More on Completely Regular Spaces
6.3.2 Characterization of Completely Regular Spaces by Real-Valued Continuous Functions
6.3.3 Tychonoff Spaces
6.4 Gδ-sets, Perfectly Normal Spaces, and Urysohn Functions
6.4.1 Gδ-sets and Fδ-sets
6.4.2 Perfectly Normal Spaces
6.4.3 Existence of Urysohn Function and Characterization of Perfectly Normal Spaces
6.5 Tietze Extension Theorem: Characterization of Normal Spaces
6.6 Rings mathcalC(X, R) for Compact Hausdorff Spaces X
6.7 The Gelfand–Kolmogoroff Theorem
6.8 Applications
6.8.1 Embedding Problems: Urysohn Metrization Theorem
6.8.2 Application to Analysis
6.8.3 Application of Baire Category Theorem
6.8.4 Extreme Value Theorem
6.8.5 Other Applications
6.9 Exercises
References
7 Countability, Separability and Embedding
7.1 Characterization of Compactness by Bolzano–Weierstrass Property
7.2 Countability and Separability
7.2.1 Lindelöf Space and Lindelöf Theorem
7.2.2 Countable Topological Spaces
7.3 Subspaces of First and Second Countable Space
7.4 Properties of Appert\'s Space and Sorgenfrey Line
7.4.1 Properties of Appert\'s Space
7.4.2 More Properties of Sorgenfrey Line
7.5 Topological Embedding Problems
7.6 More on Separability and Lindelöf Properties
7.7 Convergence of Sequences in First Countable Spaces
7.8 Cardinality of Open Sets in a Second Countable Space
7.9 Points of Condensation of Uncountable Subsets of Second Countable Spaces
7.10 More on Urysohn Metrization Theorem
7.11 Applications
7.11.1 Applications of Lindelöf Theorem
7.11.2 A Charaterization of First Countable Hausdorff Spaces
7.12 Exercises
References
8 Brief History of Topology I: Motivation of the Subject with Historical Development
8.1 Early Development of Combinatorial Topology by the Nineteenth-Century Analysts
8.2 Basic Concepts of Topology Found in the Eighteenth and the Nineteenth Centuries
8.3 Main Objective of the Study of Topology
8.4 Inauguration of General Topology
8.5 Beginning of Combinatorial Topology
8.5.1 Combined Aspect of Combinatorial and Set-Theoretic Topologies
8.5.2 Combinatorial Topology Versus General Topology
8.6 Historical Development of the Basic Topics in General Topology
8.6.1 History of Metric Spaces
8.6.2 Early Development of General Topology
8.6.3 Topological Spaces and Their Continuous Functions
8.6.4 Geometry of Continuous Mappings of Segments, Circles and Disks and Spheres
8.6.5 Product of Topology and Tychonöff Topology
8.6.6 Quotient Topology and Quotient Spaces
8.6.7 Möbius Band and Klein Bottle
8.6.8 Separation Axioms
8.6.9 Real-Valued Continuous Functions
8.6.10 Urysohn Function and Urysohn Lemma for Metric Spaces
8.6.11 Separation by Real-Valued Continuous Functions and Urysohn Lemma
8.6.12 Tietze Extension Theorem
8.6.13 Countability and Separability Axioms
8.6.14 Compactness Property
8.6.15 Stone–Čech Compactification
8.6.16 Lebesgue Lemma and Lebesgue Number
8.6.17 Paracompact Spaces
8.6.18 Baire Spaces
8.6.19 Connectedness Property
8.6.20 Compactification of Topological Spaces
8.6.21 Space-Filling Curve
8.6.22 The Gelfand–Kolmogoroff Theorem
8.6.23 Ascoli\'s Theorem
8.7 Topology of Metric Spaces
Additional Reading
Appendix Index
Index