Essentials of topology with applications:
Saved in:
| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Boca Raton [u.a.]
CRC Press
2010
|
| Series: | Textbooks in mathematics
A Chapman & Hall book |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Item Description: | Includes bibliographical references and index |
| Physical Description: | XV, 404 S. Ill., graph. Darst. |
| ISBN: | 9781420089745 |
Staff View
MARC
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| 245 | 1 | 0 | |a Essentials of topology with applications |c Steven G. Krantz |
| 264 | 1 | |a Boca Raton [u.a.] |b CRC Press |c 2010 | |
| 300 | |a XV, 404 S. |b Ill., graph. Darst. | ||
| 336 | |b txt |2 rdacontent | ||
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| 338 | |b nc |2 rdacarrier | ||
| 490 | 0 | |a Textbooks in mathematics | |
| 490 | 0 | |a A Chapman & Hall book | |
| 500 | |a Includes bibliographical references and index | ||
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| 999 | |a oai:aleph.bib-bvb.de:BVB01-018668442 | ||
Record in the Search Index
| _version_ | 1804140760089493504 |
|---|---|
| adam_text | Table
of
Contents
Preface
хш
1
Fundamentals
1
1.1
What Is Topology?
........................ 1
1.2
First Definitions
.......................... 2
1.3
Mappings
............................. 9
1.4
The Separation Axioms
...................... 12
1.5
Compactness
........................... 18
16
Homeomorphisms
......................... 24
1.7
Connectedness
........................... 28
1.8
Path-Connectedness
....................... 32
1.9
Continua
.............................. 34
1.10
Totally Disconnected Spaces
................... 36
1.11
The Cantor Set
.......................... 38
1.12
Metric Spaces
........................... 41
1.13
Metrizability
............................ 46
1.14
Baire s Theorem
.......................... 48
1.15
Lebesgue s Lemma and Lebesgue Numbers
........... 52
EXERCISES
........................... 53
2
Advanced Properties of Topological Spaces
57
2.1
Basis and Sub-Basis
....................... 57
2.2
Product Spaces
.......................... 59
2.3
Relative Topology
......................... 62
2.4
First Countable, Second Countable, and So Forth
....... 62
vii
Vlil
2.5
Compactifications
......................... 65
2.6 Quotient
Topologies
........................ 68
2.7
Uniformities
............................ 70
2.8
Morse Theory
........................... 73
2.9
Proper Mappings
......................... 79
2.10
Paracompactness
......................... 80
2.11
An Application to Digital Imaging
............... 83
EXERCISES
........................... 90
Basic Algebraic Topology
95
3.1
Homotopy Theory
......................... 95
3.2
Homology Theory
.........................108
3.2.1
Fundamentals
.......................109
3.2.2
Singular Homology
....................110
3.2.3
Relation to Homotopy
..................120
3.3
Covering Spaces
..........................127
3.4
The Concept of Index
.......................138
3.5
Mathematical Economics
.....................142
EXERCISES
...........................151
Manifold Theory
157
4.1
Basic Concepts
..........................157
4.2
The Definition
...........................159
EXERCISES
...........................166
Moore-Smith Convergence and Nets
173
5.1
Introductory Remarks
......................173
5.2
Nets
................................173
EXERCISES
...........................176
6
Function Spaces
179
6.1
Preliminary Ideas
.........................179
6.2
The Topology of Pointwise Convergence
............180
6.3
The Compact-Open Topology
..................181
їх
6.4
Uniform
Convergence
.......................182
6.5
Equicontinuity and the
Ascoli-
Arzela Theorem
.........185
6.6
The
Weierstrass
Approximation Theorem
............188
EXERCISES
...........................193
7
Knot Theory
197
7.1
What Is a Knot?
.........................197
7.2
The Alexander Polynomial
....................200
7.3
The Jones Polynomial
......................206
7.3.1
Knot Projections
.....................206
7.3.2
Reidemeister Moves
....................210
7.3.3
Bracket Polynomials
...................214
7.3.4
Creation of a New Polynomial Invariant
........216
EXERCISES
...........................220
8
Graph Theory
225
8.1
Introduction
............................225
8.2
Fundamental Ideas of Graph Theory
..............227
8.3
Application to the
Königsberg
Bridge Problem
.........231
8.4
Coloring Problems
........................235
8.4.1
Modern Developments
..................240
8.4.2
Denouement
........................242
8.5
The Traveling Salesman Problem
................242
EXERCISES
...........................245
9
Dynamical Systems
249
9.1
Flows
...............................249
9.1.1
Dynamical Systems
....................252
9.1.2
Stable and Unstable Fixed Points
............255
9.1.3
Linear Dynamics in the Plane
..............257
9.2
Planar Autonomous Systems
...................262
9.2.1
Ingredients of the Proof of
Poincaré-Bendixson
.....263
9.3
Lagrange s Equations
.......................273
EXERCISES
...........................278
Appendices
283
Appendix
1:
Principles of Logic
285
ALI
Truth
...............................286
A1.2 And and Or
.........................287
A1.3 Not
...............................290
A1.4 If- Then
............................291
Al.
5
Contrapositive,
Converse, and Iff
...............294
A1.6 Quantifiers
.............................298
Al.
7
Truth and Provability
.......................302
Appendix
2:
Principles of Set Theory
307
A2.1 Undefinable Terms
........................307
A2.2 Elements of Set Theory
......................308
A2.3 Venn Diagrams
..........................312
A2.4 Further Ideas in Elementary Set Theory
............313
A2.5 Indexing and Extended Set Operations
.............315
A2.6 Countable and Uncountable Sets
................318
Appendix
3:
The Real Numbers
333
A3.1 The Real Number System
....................333
A3.1
Construction of the Real Numbers
................339
Appendix
4:
The Axiom of Choice and Its Implications
343
A4.1 Well Ordering
...........................343
A4.2 The Continuum Hypothesis
...................344
A4.3 Zorn s Lemma
...........................344
A4.4
The Hausdorff
Maximali
ty
Principle
...............345
A4.5 The Banach-Tarski Paradox
...................345
Xl
Appendix
5:
Ideas from Algebra
347
A5.1 Groups
...............................347
A5.2 Rings
................................348
A5.3 Fields
...............................349
A5.4 Modules
..............................349
A5.5 Vector Spaces
...........................350
Solutions of Selected Exercises
351
Bibliography
389
Index
393
|
| any_adam_object | 1 |
| author | Krantz, Steven G. 1951- |
| author_GND | (DE-588)130535907 |
| author_facet | Krantz, Steven G. 1951- |
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| ctrlnum | (OCoLC)227914356 (DE-599)BVBBV035809493 |
| dewey-full | 514 |
| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 514 - Topology |
| dewey-raw | 514 |
| dewey-search | 514 |
| dewey-sort | 3514 |
| dewey-tens | 510 - Mathematics |
| discipline | Mathematik |
| format | Book |
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| indexdate | 2024-07-09T22:05:06Z |
| institution | BVB |
| isbn | 9781420089745 |
| language | English |
| lccn | 2009025306 |
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| physical | XV, 404 S. Ill., graph. Darst. |
| publishDate | 2010 |
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| publisher | CRC Press |
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| series2 | Textbooks in mathematics A Chapman & Hall book |
| spelling | Krantz, Steven G. 1951- Verfasser (DE-588)130535907 aut Essentials of topology with applications Steven G. Krantz Boca Raton [u.a.] CRC Press 2010 XV, 404 S. Ill., graph. Darst. txt rdacontent n rdamedia nc rdacarrier Textbooks in mathematics A Chapman & Hall book Includes bibliographical references and index Topology Topologie (DE-588)4060425-1 gnd rswk-swf (DE-588)4123623-3 Lehrbuch gnd-content Topologie (DE-588)4060425-1 s DE-604 Digitalisierung UB Bayreuth application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=018668442&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Krantz, Steven G. 1951- Essentials of topology with applications Topology Topologie (DE-588)4060425-1 gnd |
| subject_GND | (DE-588)4060425-1 (DE-588)4123623-3 |
| title | Essentials of topology with applications |
| title_auth | Essentials of topology with applications |
| title_exact_search | Essentials of topology with applications |
| title_full | Essentials of topology with applications Steven G. Krantz |
| title_fullStr | Essentials of topology with applications Steven G. Krantz |
| title_full_unstemmed | Essentials of topology with applications Steven G. Krantz |
| title_short | Essentials of topology with applications |
| title_sort | essentials of topology with applications |
| topic | Topology Topologie (DE-588)4060425-1 gnd |
| topic_facet | Topology Topologie Lehrbuch |
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