Undergraduate Topology

ISBN-10: 0486474194
ISBN-13: 9780486474199
Edition: 2009
List price: $20.25 Buy it from $9.84
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Description: General topology offers a valuable tool to students of mathematics, particularly in such courses as complex, real, and functional analysis. This introductory treatment is essentially self-contained and features explanations and proofs that relate to  More...

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Book details

List price: $20.25
Copyright year: 2009
Publisher: Dover Publications, Incorporated
Publication date: 10/22/2009
Binding: Paperback
Pages: 304
Size: 6.10" wide x 9.09" long x 0.71" tall
Weight: 0.792
Language: English

General topology offers a valuable tool to students of mathematics, particularly in such courses as complex, real, and functional analysis. This introductory treatment is essentially self-contained and features explanations and proofs that relate to every practical aspect of point set topology. Hundreds of exercises appear throughout the text. 1977 edition.

Sets, Functions, and Relations
Sets and Membership
Some Remarks on the Use of the Connectives and, or, implies
Subsets
Union and Intersection of Sets
Complementation
Set Identities and Other Set Relations
Counterexamples
Collections of Sets
Cartesian Product
Functions
Relations
Set Inclusions for Image and Inverse Image Seta
The Restriction of a Function
Composition of Functions
Sequences
Subsequences
Finite Induction and Well-Ordering for Positive Integers
Sequencea Defined Inductively
Some Important Properties of Relations
Decomposition of a Set
Equivalence Classes
Partially Ordered and Totally Ordered Seta
Properties of Boundedness for Partially Ordered Sets
Axiom of Choice and Zorn's Lemma
Cardinality of Sets (Introduction)
Countable Sets
Uncountable Sets
Nonequivalent Sets
Review Exercises
Structure Of R and Rn
Algebraic Structure of R
Distance Between Two Points in R
Limit of a Sequence in R
The Nested Interval Theorem for R
Algebraic Structure for R<sup>n</sup>
The Cauchy-Schwarz Inequality
The Distance Formula in R<sup>n</sup>
Open Subsets of R<sup>n</sup>
Limit Points in R<sup>n</sup>
Closed Subsets of R<sup>n</sup>
Bounded Subsets of R
Convergent Sequences in R<sup>n</sup>
Cauchy Criterion for Convergence
Some Additional Properties for R<sup>n</sup>
Some Further Remarks about R<sup>n</sup>
Metric Spaces: Introduction
Distance Function and Metric Spaces
Open Sets and Closed Sets
Some Basic Theorems Concerning Open and Closed Sets
Topology Generated by a Metric
Subspace of a Metric Space
Convergent Sequences in Metric Spaces
Cartesian Product of a Finite Number of Metric Spaces
Continuous Mappings: Introduction
Uniform Continuity
Metric Spaces: Special Properties and Mappings on Metric Spaces
Separation Properties
Connectedness in Metric Spaces
The Invariance of Connectedness under Continuous Mappings
Polygonal Connectedness
Separable Metric Spaces
Totally Bounded Metric Spaces
Sequential Compactness for Metric Spaces
The Bolzano-Weierstrass Property
Compactness or Finite Subcovering Property
Complete Metric Spaces
Nested Sequences of Sets for Complete Spaces
Another Characterization of Compact Metric Spaces
Completion of a Metric Space
Sequences of Mappings into a Metric Space
Review Exercises
Metric Spaces: Some Examples and Applications
Linear or Vector Spaces
The Hilbert Space l<sup>2</sup>
The Hilbert Cube
The Space & ([a, b]) of Continuous Real-Valued Mappings on a Closed Interval [a, b]
An Application of Completeness: Contraction Mappings
Fundamental Existence Theorem for First Order Differential Equations-An Application of the Banach Fixed Point Theorem
General Topological Spaces and Mappings on Topological Spaces
Topological Spaces
Base for a Topology
Some Basic Definitions
Some Basic Theorems for Topological Spaces
Neighborhoods and Neighborhood Systems
Subspaces
Continuous and Topological Mappings
Some Basic Theorems Concerning Mappings
Separation Properties for Topological Spaces
A Characterization of Normality
Separability Axiom
Second Countable Spaces
First Countable Spaces
Comparison of Topologies
Urysohn's Metrization Theorem
Compactness And Related Properties
Definitions of Various Compactness Properties
Some Consequences of Compactness
Relations Between Various Types of Compactness
Local Compactness
The One-Point Compactification
Some Generalizations of Mappings Denned on Compact Spaces
Connectedness And Related Concepts
Connectedness. Definitions
Some Basic Theorems Concerning Connectedness
Limit Superior and Limit Inferior of Sequences of Subsets of a Space
Review Exercises
Quotient Spaces
Decomposition of a Topological Space
Quasi-Compact Mappings
The Quotient Topology
Decomposition of a Domain Space into Point Inverses
Topologically Equivalent Mappings
Decomposition of a Domain Space into Components of Point Inverses
Factorization of Compact Mappings
Net and Filter Convergence
Nets and Subnets
Convergence of Nets
Filters
Product Spaces
Cartesian Products
The Product Topology
Mappings into Product Spaces
References
Index

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