Topological Methods in Euclidean Spaces, Extensive development of

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Preface ix
Chapter 1 Point-set topology of Euclidean spaces 1
1 Introduction 1
2 Preliminaries 6
3 Open sets, closed sets, and continuity 10
4 Compact spaces 21
5 Connectivity properties 25
6 Real-valued continuous functions 34
7 Retracts 38
8 Topological dimension 41
Supplementary exercises 44
Chapter 2 Elementary combinatorial techniques 46
1 Introduction 46
2 Hyperplanes in R[superscript n] 46
3 Simplexes and complexes 49
4 Sample triangulations 55
5 Simplicial maps 60
6 Barycentric subdivision 63
7 The Simplicial Approximation Theorem 70
8 Sperner's Lemma 73
9 The Brouwer Fixed Point Theorem 75
10 Topological dimension of compact subsets of R[superscript n] 77
Supplementary exercises 79
Chapter 3 Homotopy theory and the fundamental group 81
1 Introduction 81
2 The homotopy relation, nullhomotopic maps, and contractible spaces 84
3 Maps of spheres 86
4 The fundamental group 90
5 Fundamental groups of the spheres 99
Supplementary exercises 106
Chapter 4 Simplicial homology theory 108
1 Introduction 108
2 Oriented complexes and chains 111
3 Boundary operators 116
4 Cycles, boundaries, and homology groups 118
5 Elementary examples 122
6 Cone complexes, augmented complexes, and the homology groups 125
7 Incidence numbers and the homology groups 128
8 Elementary homological algebra 130
9 The homology complex of a geometric complex 133
10 Acyclic carrier functions 136
11 Invariance of homology groups under barycentric subdivision 138
12 Homomorphisms induced by continuous maps 141
13 Homology groups of topological polyhedra 144
14 The Hopf Trace Theorem 147
15 The Lefschetz Fixed Point Theorem 150
Supplementary exercises 150
Chapter 5 Differential techniques 154
1 Introduction 154
2 Smooth maps 161
3 The Stone--Weierstrass Theorem 163
4 Derivatives as linear transformations 166
5 Differentiable manifolds 173
6 Tangent spaces and derivatives 175
7 Regular and critical values of smooth maps 179
8 Measure zero and Sard's Theorem 183
9 Morse functions 190
10 Manifolds with boundary 201
11 One-dimensional manifolds 205
12 Topological characterization of S[superscript k] 208
13 Smooth tangent vector fields 211
Supplementary exercises 217
Solutions to Selected Exercises 220
Guide to further study 238
Bibliography 240
List of symbols and notation 242
Index 246

Title: Topological Methods in Euclidean Spaces

Manufacturer:

Dover Publications

Item Number: 9780486414522

Publication Date: November 2000

Number: 1

Product Description: Topological Methods in Euclidean Spaces

Universal Product Code (UPC): 9780486414522

WonderClub Stock Keeping Unit (WSKU): 9780486414522

Rating: 3/5 based on 2 Reviews

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Weight: 0.200 kg (0.44 lbs)

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Heigh : 8.600 cm (3.39 inches)

Depth: 0.500 cm (0.20 inches)

Date Added: August 25, 2020, Added By: Ross

Date Last Edited: August 25, 2020, Edited By: Ross

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Scott Henson

reviewed Topological Methods in Euclidean Spaces on May 28, 2017

Topological Methods in Euclidean Spaces

La prima cosa da sapere su questo libro Ã¨ che PÃ©ter Ã¨ il cognome di una matematica ungherese del '900, che oltre ai suoi lavori accademici scrisse questo testo di introduzione ai temi matematici che ebbe un grande successo in tutto il mondo, con traduzioni in svariate lingue tra cui per l'appunto l'italiano. Onestamente ho trovato l'approccio seguito un po' pesante per i miei gusti, un po' per la traduzione di un allora giovane Giulio Giorello che seguiva chiaramente lo stile dell'epoca, ma soprattutto perchÃ© il testo mi ha dato l'idea di sfruttare sempre casi particolari per giungere a una generalizzazione che perÃ² non ha un vero fondamento. La cosa si vede molto bene nella parte iniziale, dove si comincia con le operazioni elementari: certo che andando avanti fino ai teoremi di indeterminazione di GÃ¶del e all'induzione transfinita di Gentzen (sÃ¬, PÃ©ter arriva sin lÃ , cosa che negli anni '50 del secolo scorso non era per nulla scontata) le cose migliorano. Ho qualche dubio sull'approccio che dai disegnini discreti porta allo studio delle curve, ma lÃ¬ il problema potrebbe essere mio.
Un'ultima nota. Io ho letto la prima edizione italiana del libro, pubblicata da Feltrinelli nel 1973, e nella quale c'erano alcuni errori nel testo (anche stupidi, come scrivere 2/6 e 2/10 anzichÃ© 6/2 e 10/2). Per curiositÃ ho dato un'occhiata alle versione attuale, edita nel 2010 da Rizzoli, e almeno quell'errore Ã¨ rimasto intatto. Ci deve essere stata una grande fiducia nella traduzione originale, insomma!

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	Preface	ix
Chapter 1	Point-set topology of Euclidean spaces	1
1	Introduction	1
2	Preliminaries	6
3	Open sets, closed sets, and continuity	10
4	Compact spaces	21
5	Connectivity properties	25
6	Real-valued continuous functions	34
7	Retracts	38
8	Topological dimension	41
	Supplementary exercises	44
Chapter 2	Elementary combinatorial techniques	46
1	Introduction	46
2	Hyperplanes in R[superscript n]	46
3	Simplexes and complexes	49
4	Sample triangulations	55
5	Simplicial maps	60
6	Barycentric subdivision	63
7	The Simplicial Approximation Theorem	70
8	Sperner's Lemma	73
9	The Brouwer Fixed Point Theorem	75
10	Topological dimension of compact subsets of R[superscript n]	77
	Supplementary exercises	79
Chapter 3	Homotopy theory and the fundamental group	81
1	Introduction	81
2	The homotopy relation, nullhomotopic maps, and contractible spaces	84
3	Maps of spheres	86
4	The fundamental group	90
5	Fundamental groups of the spheres	99
	Supplementary exercises	106
Chapter 4	Simplicial homology theory	108
1	Introduction	108
2	Oriented complexes and chains	111
3	Boundary operators	116
4	Cycles, boundaries, and homology groups	118
5	Elementary examples	122
6	Cone complexes, augmented complexes, and the homology groups	125
7	Incidence numbers and the homology groups	128
8	Elementary homological algebra	130
9	The homology complex of a geometric complex	133
10	Acyclic carrier functions	136
11	Invariance of homology groups under barycentric subdivision	138
12	Homomorphisms induced by continuous maps	141
13	Homology groups of topological polyhedra	144
14	The Hopf Trace Theorem	147
15	The Lefschetz Fixed Point Theorem	150
	Supplementary exercises	150
Chapter 5	Differential techniques	154
1	Introduction	154
2	Smooth maps	161
3	The Stone--Weierstrass Theorem	163
4	Derivatives as linear transformations	166
5	Differentiable manifolds	173
6	Tangent spaces and derivatives	175
7	Regular and critical values of smooth maps	179
8	Measure zero and Sard's Theorem	183
9	Morse functions	190
10	Manifolds with boundary	201
11	One-dimensional manifolds	205
12	Topological characterization of S[superscript k]	208
13	Smooth tangent vector fields	211
	Supplementary exercises	217
	Solutions to Selected Exercises	220
	Guide to further study	238
	Bibliography	240
	List of symbols and notation	242
	Index	246