Elementary point-set topology : a transition to advanced mathematics / André L. Yandl, Seattle University, Adam Bowers, University of California San Diego.

Format:

Book

Author/Creator:

Yandl, André L., author.

Bowers, Adam (Adam Roman), author.

Series:

Aurora: Dover modern math originals

Dover books on mathematics

Aurora Dover modern math originals

Language:

English

Subjects (All):

Point set theory.

Propositional calculus.

Topology.

Calculus.

Physical Description:

254 pages ; 23 cm.

Place of Publication:

Mineola, New York : Dover Publications, Inc., 2016.

Summary:

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. Book jacket.

Contents:

1 Mathematical Proofs and Sets 13

1.1 Introduction to Elementary Logic 13

1.2 More Elementary Logic 17

1.3 Quantifiers 23

1.4 Methods of Mathematical Proof 29

1.5 Introduction to Elementary Set Theory 40

1.6 Cardinality 53

1.7 Cardinal Arithmetic 61

2 Topological Spaces 67

2.1 Introduction 67

2.2 Topologies 69

2.3 Bases 74

2.4 Subspaces 79

2.5 Interior, Closure, and Boundary 82

2.6 Hausdorff spaces 92

2.7 Metric Spaces 96

2.8 Euclidean Spaces 105

3 Continuous Functions 111

3.1 Review of the Function Concept 111

3.2 More on Image and Inverse Image 118

3.3 Continuous Functions 124

3.4 More on Continuous Functions 133

3.5 More on Homeomorphism 142

4 Product Spaces 149

4.1 Products of Sets 149

4.2 Product Spaces 156

4.3 More on Product Spaces 162

5 Connectedness 167

5.1 Introduction to Connectedness 167

5.2 Products of Connected Spaces 173

5.3 Connected Subsets of the Real Line 176

6 Compactness 185

6.1 Introduction to Compactness 185

6.2 Compactness in the Space of Real Numbers 191

6.3 The Product of Compact Spaces 194

6.4 Compactness in Metric Spaces 198

6.5 More on Compactness in Metric Spaces 205

6.6 The Cantor Set 210

7 Fixed Point Theorems and Applications 217

7.1 Sperner's Lemma 217

7.2 Brouwer's Fixed Point Theorem 221

7.3 The Fundamental Theorem of Algebra 226

7.4 Function Spaces 233

7.5 Contractions 239.

Notes:

Includes index.

ISBN:

9780486803494

048680349X

OCLC:

909082940

Publisher Number:

99968461490

Online:

Contributor biographical information

Publisher description