Elementary Point-Set Topology A Transition to Advanced Mathematics

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  • Format: Paperback
  • Copyright: 2016-05-18
  • Publisher: Dover Publications

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In addition to serving as an introduction to the basics of point-set topology, this text bridges the gap between the elementary calculus sequence and higher-level mathematics courses. The versatile, original approach focuses on learning to read and write proofs rather than covering advanced topics. Based on lecture notes that were developed over many years at The University of Seattle, the treatment is geared toward undergraduate math majors and suitable for a variety of introductory courses.
Starting with elementary concepts in logic and basic techniques of proof writing, the text defines topological and metric spaces and surveys continuity and homeomorphism. Additional subjects include product spaces, connectedness, and compactness. The final chapter illustrates topology's use in other branches of mathematics with proofs of the fundamental theorem of algebra and of Picard's existence theorem for differential equations.
"This is a back-to-basics introductory text in point-set topology that can double as a transition to proofs course. The writing is very clear, not too concise or too wordy. Each section of the book ends with a large number of exercises. The optional first chapter covers set theory and proof methods; if the students already know this material you can start with Chapter 2 to present a straight topology course, otherwise the book can be used as an introduction to proofs course also." — Mathematical Association of America

Author Biography

André L. Yandl is Professor Emeritus of Mathematics at the University of Seattle.
Adam Bowers is a Lecturer in Mathematics at the University of California, San Diego.

Table of Contents

List of Figures
List of Symbols
1 Mathematical Proofs and Sets
1.1 Introduction to Elementary Logic .
1.2 More Elementary Logic
1.3 Quantifiers
1.4 Methods ofMathematical Proof
1.5 Introduction to Elementary Set Theory
1.6 Cardinality
1.7 Cardinal Arithmetic
2 Topological Spaces
2.1 Introduction
2.2 Topologies
2.3 Bases
2.4 Subspaces
2.5 Interior, Closure, and Boundary
2.6 Hausdorff spaces
2.7 Metric Spaces
2.8 Euclidean Spaces
3 Continuous Functions
3.1 Review of the Function Concept
3.2 More on Image and Inverse Image
3.3 Continuous Functions
3.4 More on Continuous Functions
3.5 More on Homeomorphism
4 Product Spaces
4.1 Products of Sets
4.2 Product Spaces
4.3 More on Product Spaces
5 Connectedness
5.1 Introduction to Connectedness
5.2 Products of Connected Spaces
5.3 Connected Subsets of the Real Line
6 Compactness
6.1 Introduction to Compactness
6.2 Compactness in the Space of Real Numbers
6.3 The Product of Compact Spaces
6.4 Compactness in Metric Spaces
6.5 More on Compactness in Metric Spaces
6.6 The Cantor Set
7 Fixed Point Theorems and Applications
7.1 Sperner’s
7.2 Brouwer’s Fixed Point Theorem.
7.3 The Fundametnal Theorem of Algebra
7.4 Function Spaces
7.5 Contractions

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