Counterexamples in topology:
Gespeichert in:
Hauptverfasser: | , |
---|---|
Format: | Buch |
Sprache: | English |
Veröffentlicht: |
New York
Dover Publ.
1995
|
Ausgabe: | [Reprint], unabridged and unaltered republ. of the 2. ed., 1978 |
Schlagworte: | |
Online-Zugang: | Inhaltsverzeichnis |
Beschreibung: | XI, 244 S. Ill., graph. Darst. |
ISBN: | 048668735X |
Internformat
MARC
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300 | |a XI, 244 S. |b Ill., graph. Darst. | ||
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Datensatz im Suchindex
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adam_text | Contents
Part I BASIC DEFINITIONS
1· General Introduction 3
Limit Points 5
Closures and Interiors 6
Countability Properties 7
Functions 7
Filters 9
2. Separation Axioms 11
Regular and Normal Spaces 12
Completely Hausdorff Spaces 13
Completely Regular Spaces 13
Functions, Products, and Subspaces 14
Additional Separation Properties 16
3. Compactness 18
Global Compactness Properties 18
Localized Compactness Properties 20
Countability Axioms and Separability 21
Paracompactness 22
Compactness Properties and Ti Axioms 24
Invariance Properties 26
4. Connectedness 28
Functions and Products 31
Disconnectedness 31
Biconnectedness and Continua 33
vii
viii Contents
5. Metric Spaces 34
Complete Metric Spaces 36
Metrizability 37
Uniformities 37
Metric Uniformities 38
Partii COUNTEREXAMPLES
1. Finite Discrete Topology 41
2. Countable Discrete Topology 41
3. Uncountable Discrete Topology 41
4. Indiscrete Topology 42
5. Partition Topology 43
6. Odd-Even Topology 43
7. Deleted Integer Topology 43
8. Finite Particular Point Topology 44
9. Countable Particular Point Topology 44
10. Uncountable Particular Point Topology 44
11. Sierpiński Space 44
12. Closed Extension Topology 44
13. Finite Excluded Point Topology 47
14. Countable Excluded Point Topology 47
15. Uncountable Excluded Point Topology 47
16. Open Extension Topology 47
17. Either-Or Topology 48
18. Finite Complement Topology on a Countable Space 49
19. Finite Complement Topology on an Uncountable Space 49
20. Countable Complement Topology 50
21. Double Pointed Countable Complement Topology 50
22. Compact Complement Topology 51
23. Countable Fort Space 52
24. Uncountable Fort Space 52
25. Fortissimo Space 53
26. Arens-Fort Space 54
27. Modified Fort Space 55
28. Euclidean Topology 56
29. The Cantor Set 57
30. The Rational Numbers 59
31. The Irrational Numbers 59
32. Special Subsets of the Real Line 60
33. Special Subsets of the Plane 61
34. One Point Compactification Topology 63
Contents ix
35. One Point Compactification of the Rationals 63
36. Hilbert Space 64
37. Fréchet Space 64
38. Hilbert Cube 65
39. Order Topology 66
40. Open Ordinal Space [0,P) (T Ù) 68
4R Closed Ordinal Space [0,r] (V ü) 68
42. Open Ordinal Space [0,Q) 68
43. Closed Ordinal Space [0,0] 68
44. Uncountable Discrete Ordinal Space 70
45. The Long Line 71
46. The Extended Long Line 71
47. An Altered Long Line 72
48. Lexicographic Ordering on the Unit Square 73
49. Right Order Topology 74
50. Right Order Topology on R 74
51. Right Half-Open interval Topology 75
52. Nested Interval Topology 76
53. Overlapping Interval Topology 77
54. Interlocking Interval Topology 77
55. Hjalmar Ekdal Topology 78
56. Prime Ideal Topology 79
57. Divisor Topology 79
58. Evenly Spaced Integer Topology 80
59. The p-adic Topology on Z 81
60. Relatively Prime Integer Topology 82
61. Prime Integer Topology 82
62. Double Pointed Reals 84
63. Countable Complement Extension Topology 85
64. Smirnovas Deleted Sequence Topology 86
65. Rational Sequence Topology 87
66. Indiscrete Rational Extension of R 88
67. Indiscrete Irrational Extension of R 88
68. Pointed Rational Extension of R 88
69. Pointed Irrational Extension of R 88
70. Discrete Rational Extension of R 90
71. Discrete Irrational Extension of R 90
72. Rational Extension in the Plane 91
73. Telophase Topology 92
74. Double Origin Topology 92
75. Irrational Slope Topology 93
76. Deleted Diameter Topology 94
Contents
77. Deleted Radius Topology 94
78. Half-Disc Topology 96
79. Irregular Lattice Topology 97
80. Arens Square 98
81. Simplified Arens Square 100
82. Niemytzki’s Tangent Disc Topology 100
83. Metrizable Tangent Disc Topology 103
84. Sorgenfrey’s Half-Open Square Topology 103
85. Michael s Product Topology 105
86. Tychonoff Plank 106
87. Deleted Tychonoff Plank 106
88. Alexandroff Plank 107
89. Dieudonne Plank 108
90. Tychonoff Corkscrew 109
91. Deleted Tychonoff Corkscrew 109
92. Hewitt s Condensed Corkscrew 111
93. Thomas’ Plank 113
94. Thomas’ Corkscrew 113
95. Weak Parallel Line Topology 114
96. Strong Parallel Line Topology 114
97. Concentric Circles 116
98. Appert Space 117
99. Maximal Compact Topology 118
100. Minimal Hausdorff Topology 119
101. Alexandroff Square 120
102. Zz 121
103. Uncountable Products of Z+ 123
104. Baire Product Metric on Rw 124
105. P 125
106. [0,0) X P 126
107. Helly Space 127
108. C[0,1] 128
109. Box Product Topology on 128
110. Stone-Cech Compactification 129
111. Stone-Cech Compactification of the Integers 132
112. Novak Space 134
113. Strong Ultrafilter Topology 135
114. Single Ultrafilter Topology 136
115. Nested Rectangles 137
116. Topologist’s Sine Curve 137
117. Closed Topologist’s Sine Curve 137
118. Extended Topologist’s Sine Curve 137
119. The Infinite Broom 139
120. The Closed Infinite Broom 139
121. The Integer Broom 140
122. Nested Angles 140
123. The Infinite Cage 141
124. Bernstein’s Connected Sets 142
125. Gustin’s Sequence Space 142
126. Roy’s Lattice Space 143
127. Roy’s Lattice Subspace 143
128. Cantor’s Leaky Tent 145
129. Cantor’s Teepee 145
130. A Pseudo-Arc 147
131. Miller’s Biconnected Set 148
132. Wheel without Its Hub 150
133. Tangora’s Connected Space 150
134. Bounded Metrics 151
135. Sierpiński’s Metric Space 152
136. Duncan’s Space 153
137. Cauchy Completion 154
138. Hausdorff’s Metric Topology 154
139. The Post Office Metric 155
140. The Radial Metric 155
141. Radial Interval Topology 156
142. Bing’s Discrete Extension Space 157
143. Michael’s Closed Subspace 157
Part III METRIZATION THEORY
Conjectures and Counterexamples 161
Part IV APPENDICES
Special Reference Charts 185
Separation Axiom Chart 187
Compactness Chart 188
Paracompactness Chart 190
Connectedness Chart 191
Disconnectedness Chart 192
Metrizability Chart 193
General Reference Chart 195
Problems 205
Notes 213
Bibliography 228
Index 236
|
adam_txt |
Contents
Part I BASIC DEFINITIONS
1· General Introduction 3
Limit Points 5
Closures and Interiors 6
Countability Properties 7
Functions 7
Filters 9
2. Separation Axioms 11
Regular and Normal Spaces 12
Completely Hausdorff Spaces 13
Completely Regular Spaces 13
Functions, Products, and Subspaces 14
Additional Separation Properties 16
3. Compactness 18
Global Compactness Properties 18
Localized Compactness Properties 20
Countability Axioms and Separability 21
Paracompactness 22
Compactness Properties and Ti Axioms 24
Invariance Properties 26
4. Connectedness 28
Functions and Products 31
Disconnectedness 31
Biconnectedness and Continua 33
vii
viii Contents
5. Metric Spaces 34
Complete Metric Spaces 36
Metrizability 37
Uniformities 37
Metric Uniformities 38
Partii COUNTEREXAMPLES
1. Finite Discrete Topology 41
2. Countable Discrete Topology 41
3. Uncountable Discrete Topology 41
4. Indiscrete Topology 42
5. Partition Topology 43
6. Odd-Even Topology 43
7. Deleted Integer Topology 43
8. Finite Particular Point Topology 44
9. Countable Particular Point Topology 44
10. Uncountable Particular Point Topology 44
11. Sierpiński Space 44
12. Closed Extension Topology 44
13. Finite Excluded Point Topology 47
14. Countable Excluded Point Topology 47
15. Uncountable Excluded Point Topology 47
16. Open Extension Topology 47
17. Either-Or Topology 48
18. Finite Complement Topology on a Countable Space 49
19. Finite Complement Topology on an Uncountable Space 49
20. Countable Complement Topology 50
21. Double Pointed Countable Complement Topology 50
22. Compact Complement Topology 51
23. Countable Fort Space 52
24. Uncountable Fort Space 52
25. Fortissimo Space 53
26. Arens-Fort Space 54
27. Modified Fort Space 55
28. Euclidean Topology 56
29. The Cantor Set 57
30. The Rational Numbers 59
31. The Irrational Numbers 59
32. Special Subsets of the Real Line 60
33. Special Subsets of the Plane 61
34. One Point Compactification Topology 63
Contents ix
35. One Point Compactification of the Rationals 63
36. Hilbert Space 64
37. Fréchet Space 64
38. Hilbert Cube 65
39. Order Topology 66
40. Open Ordinal Space [0,P) (T Ù) 68
4R Closed Ordinal Space [0,r] (V ü) 68
42. Open Ordinal Space [0,Q) 68
43. Closed Ordinal Space [0,0] 68
44. Uncountable Discrete Ordinal Space 70
45. The Long Line 71
46. The Extended Long Line 71
47. An Altered Long Line 72
48. Lexicographic Ordering on the Unit Square 73
49. Right Order Topology 74
50. Right Order Topology on R 74
51. Right Half-Open interval Topology 75
52. Nested Interval Topology 76
53. Overlapping Interval Topology 77
54. Interlocking Interval Topology 77
55. Hjalmar Ekdal Topology 78
56. Prime Ideal Topology 79
57. Divisor Topology 79
58. Evenly Spaced Integer Topology 80
59. The p-adic Topology on Z 81
60. Relatively Prime Integer Topology 82
61. Prime Integer Topology 82
62. Double Pointed Reals 84
63. Countable Complement Extension Topology 85
64. Smirnovas Deleted Sequence Topology 86
65. Rational Sequence Topology 87
66. Indiscrete Rational Extension of R 88
67. Indiscrete Irrational Extension of R 88
68. Pointed Rational Extension of R 88
69. Pointed Irrational Extension of R 88
70. Discrete Rational Extension of R 90
71. Discrete Irrational Extension of R 90
72. Rational Extension in the Plane 91
73. Telophase Topology 92
74. Double Origin Topology 92
75. Irrational Slope Topology 93
76. Deleted Diameter Topology 94
Contents
77. Deleted Radius Topology 94
78. Half-Disc Topology 96
79. Irregular Lattice Topology 97
80. Arens Square 98
81. Simplified Arens Square 100
82. Niemytzki’s Tangent Disc Topology 100
83. Metrizable Tangent Disc Topology 103
84. Sorgenfrey’s Half-Open Square Topology 103
85. Michael's Product Topology 105
86. Tychonoff Plank 106
87. Deleted Tychonoff Plank 106
88. Alexandroff Plank 107
89. Dieudonne Plank 108
90. Tychonoff Corkscrew 109
91. Deleted Tychonoff Corkscrew 109
92. Hewitt's Condensed Corkscrew 111
93. Thomas’ Plank 113
94. Thomas’ Corkscrew 113
95. Weak Parallel Line Topology 114
96. Strong Parallel Line Topology 114
97. Concentric Circles 116
98. Appert Space 117
99. Maximal Compact Topology 118
100. Minimal Hausdorff Topology 119
101. Alexandroff Square 120
102. Zz 121
103. Uncountable Products of Z+ 123
104. Baire Product Metric on Rw 124
105. P 125
106. [0,0) X P 126
107. Helly Space 127
108. C[0,1] 128
109. Box Product Topology on 128
110. Stone-Cech Compactification 129
111. Stone-Cech Compactification of the Integers 132
112. Novak Space 134
113. Strong Ultrafilter Topology 135
114. Single Ultrafilter Topology 136
115. Nested Rectangles 137
116. Topologist’s Sine Curve 137
117. Closed Topologist’s Sine Curve 137
118. Extended Topologist’s Sine Curve 137
119. The Infinite Broom 139
120. The Closed Infinite Broom 139
121. The Integer Broom 140
122. Nested Angles 140
123. The Infinite Cage 141
124. Bernstein’s Connected Sets 142
125. Gustin’s Sequence Space 142
126. Roy’s Lattice Space 143
127. Roy’s Lattice Subspace 143
128. Cantor’s Leaky Tent 145
129. Cantor’s Teepee 145
130. A Pseudo-Arc 147
131. Miller’s Biconnected Set 148
132. Wheel without Its Hub 150
133. Tangora’s Connected Space 150
134. Bounded Metrics 151
135. Sierpiński’s Metric Space 152
136. Duncan’s Space 153
137. Cauchy Completion 154
138. Hausdorff’s Metric Topology 154
139. The Post Office Metric 155
140. The Radial Metric 155
141. Radial Interval Topology 156
142. Bing’s Discrete Extension Space 157
143. Michael’s Closed Subspace 157
Part III METRIZATION THEORY
Conjectures and Counterexamples 161
Part IV APPENDICES
Special Reference Charts 185
Separation Axiom Chart 187
Compactness Chart 188
Paracompactness Chart 190
Connectedness Chart 191
Disconnectedness Chart 192
Metrizability Chart 193
General Reference Chart 195
Problems 205
Notes 213
Bibliography 228
Index 236 |
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discipline_str_mv | Mathematik |
edition | [Reprint], unabridged and unaltered republ. of the 2. ed., 1978 |
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id | DE-604.BV021268461 |
illustrated | Illustrated |
index_date | 2024-07-02T13:43:49Z |
indexdate | 2024-07-09T20:34:17Z |
institution | BVB |
isbn | 048668735X |
language | English |
oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-014589619 |
oclc_num | 474315528 |
open_access_boolean | |
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owner_facet | DE-19 DE-BY-UBM DE-20 DE-11 DE-83 DE-523 DE-739 DE-92 DE-29T |
physical | XI, 244 S. Ill., graph. Darst. |
publishDate | 1995 |
publishDateSearch | 1995 |
publishDateSort | 1995 |
publisher | Dover Publ. |
record_format | marc |
spelling | Steen, Lynn Arthur Verfasser aut Counterexamples in topology Lynn Arthur Steen and J. Arthur Seebach [Reprint], unabridged and unaltered republ. of the 2. ed., 1978 New York Dover Publ. 1995 XI, 244 S. Ill., graph. Darst. txt rdacontent n rdamedia nc rdacarrier Topologie (DE-588)4060425-1 gnd rswk-swf Topologischer Raum (DE-588)4137586-5 gnd rswk-swf Gegenbeispiel (DE-588)4214218-0 gnd rswk-swf Topologie (DE-588)4060425-1 s Gegenbeispiel (DE-588)4214218-0 s DE-604 Topologischer Raum (DE-588)4137586-5 s 1\p DE-604 Seebach, J. Arthur Verfasser aut Digitalisierung UB Passau - ADAM Catalogue Enrichment application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=014589619&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis 1\p cgwrk 20201028 DE-101 https://d-nb.info/provenance/plan#cgwrk |
spellingShingle | Steen, Lynn Arthur Seebach, J. Arthur Counterexamples in topology Topologie (DE-588)4060425-1 gnd Topologischer Raum (DE-588)4137586-5 gnd Gegenbeispiel (DE-588)4214218-0 gnd |
subject_GND | (DE-588)4060425-1 (DE-588)4137586-5 (DE-588)4214218-0 |
title | Counterexamples in topology |
title_auth | Counterexamples in topology |
title_exact_search | Counterexamples in topology |
title_exact_search_txtP | Counterexamples in topology |
title_full | Counterexamples in topology Lynn Arthur Steen and J. Arthur Seebach |
title_fullStr | Counterexamples in topology Lynn Arthur Steen and J. Arthur Seebach |
title_full_unstemmed | Counterexamples in topology Lynn Arthur Steen and J. Arthur Seebach |
title_short | Counterexamples in topology |
title_sort | counterexamples in topology |
topic | Topologie (DE-588)4060425-1 gnd Topologischer Raum (DE-588)4137586-5 gnd Gegenbeispiel (DE-588)4214218-0 gnd |
topic_facet | Topologie Topologischer Raum Gegenbeispiel |
url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=014589619&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
work_keys_str_mv | AT steenlynnarthur counterexamplesintopology AT seebachjarthur counterexamplesintopology |