Taste of Topology

ISBN-10: 038725790X
ISBN-13: 9780387257907
Edition: 2005
Authors: Volker Runde
List price: $49.95 Buy it from $31.12
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Description: If mathematics is a language, then taking a topology course at the undergraduate level is cramming vocabulary and memorizing irregular verbs: a necessary, but not always exciting exercise one has to go through before one can read great works of  More...

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Book details

List price: $49.95
Copyright year: 2005
Publisher: Springer
Publication date: 12/7/2007
Binding: Paperback
Pages: 182
Size: 6.00" wide x 9.00" long x 0.50" tall
Weight: 0.594
Language: English

If mathematics is a language, then taking a topology course at the undergraduate level is cramming vocabulary and memorizing irregular verbs: a necessary, but not always exciting exercise one has to go through before one can read great works of literature in the original language. The present book grew out of notes for an introductory topology course at the University of Alberta. It provides a concise introduction to set theoretic topology (and to a tiny little bit of algebraic topology). It is accessible to undergraduates from the second year on, but even beginning graduate students canbenefit from some parts. Great care has been devoted to the selection of examples that are not self-serving, but already accessible for students who have a background in calculus and elementary algebra, but not necessarily in real or complex analysis. In some points, the book treats its material differently than other texts on the subject: * Baire's theorem is derived from Bourbaki's Mittag-Leffler theorem; * nets are used extensively, in particular for an intuitive proof of Tychonoff's theorem; * a short and elegant, but little known proof for the Stone-Weierstrass theorem is given.

Preface
Introduction
Set Theory
Sets and Functions
Cardinals
Cartesian Products
Remarks
Metric Spaces
Definitions and Examples
Open and Closed Sets
Convergence and Continuity
Completeness
Compactness for Metric Spaces
Remarks
Set-Theoretic Topology
Topological Spaces-Definitions and Examples
Continuity and Convergence of Nets
Compactness
Connectedness
Separation Properties
Remarks
Systems of Continuous Functions
Urysohn's Lemma and Applications
The Stone-Čech Compactification
The Stone-Weierstra� Theorems
Remarks
Basic Algebraic Topology
Homotopy and the Fundamental Group
Covering Spaces
Remarks
The Classical Mittag-Leffler Theorem Derived from Bourbaki's
Failure of the Heine-Borel Theorem in Infinite-Dimensional Spaces
The Arzel�-Ascoli Theorem
References
Index

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