Counterexamples in Topology

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Springer Science & Business Media, Dec 6, 2012 - Mathematics - 244 pages
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The creative process of mathematics, both historically and individually, may be described as a counterpoint between theorems and examples. Al though it would be hazardous to claim that the creation of significant examples is less demanding than the development of theory, we have dis covered that focusing on examples is a particularly expeditious means of involving undergraduate mathematics students in actual research. Not only are examples more concrete than theorems-and thus more accessible-but they cut across individual theories and make it both appropriate and neces sary for the student to explore the entire literature in journals as well as texts. Indeed, much of the content of this book was first outlined by under graduate research teams working with the authors at Saint Olaf College during the summers of 1967 and 1968. In compiling and editing material for this book, both the authors and their undergraduate assistants realized a substantial increment in topologi cal insight as a direct result of chasing through details of each example. We hope our readers will have a similar experience. Each of the 143 examples in this book provides innumerable concrete illustrations of definitions, theo rems, and general methods of proof. There is no better way, for instance, to learn what the definition of metacompactness really means than to try to prove that Niemytzki's tangent disc topology is not metacompact. The search for counterexamples is as lively and creative an activity as can be found in mathematics research.
 

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Contents

General Introduction
3
Separation Axioms
11
Connectedness
28
Finite Discrete Topology
41
Finite Excluded Point Topology
47
Fortissimo Space
53
The Rational Numbers
59
One Point Compactification of the Rationals
63
Concentric Circles
116
Appert Space
117
Maximal Compact Topology
118
Minimal Hausdorff Topology
119
Alexandroff Square
120
ZZ
121
Uncountable Products of Z+
123
Baire Product Metric on R
124

Hilbert Space
64
Hilbert Cube
65
Order Topology
66
Open Ordinal Space 0I T Q
68
Uncountable Piscrete Ordinal Space
70
The Long Line
71
An Altered Long Line
72
Lexicographic Ordering on the Unit Square
73
Right Order Topology
74
Right HalfOpen Interval Topology
75
Nested Interval Topology
76
Overlapping Interval Topology
77
Hjalmar Ekdal Topology
78
Prime Ideal Topology
79
Evenly Spaced Integer Topology
80
The padic Topology on Z
81
Relatively Prime Integer Topology
82
Double Pointed Reals
84
Countable Complement Extension Topology
85
Smirnovs Deleted Sequence Topology
86
Rational Sequence Topology
87
Indiscrete Rational Extension of R
88
Discrete Rational Extension of R
90
Rational Extension in the Plane
91
Telophase Topology
92
Irrational Slope Topology
93
Deleted Diameter Topology
94
HalfDisc Topology
96
Irregular Lattice Topology
97
Arens Square
98
Simplified Arens Square
100
Metrizable Tangent Disc Topology
103
Michaels Product Topology
105
Tychonoff Plank
106
Alexandroff Plank
107
Dieudonne Plank
108
Tychonoff Corkscrew
109
Hewitts Condensed Corkscrew
111
Thomas Plank
113
Weak Parallel Line Topology
114
00 X II 126 107 Helly Space
127
C01 128 109 Box Product Topology on R
128
StoneČech Compactification
129
StoneČech Compactification of the Integers
132
Novak Space
134
Strong Ultrafilter Topology
135
Single Ultrafilter Topology
136
Nested Rectangles
137
The Infinite Broom
139
The Integer Broom
140
The Infinite Cage
141
Bernsteins Connected Sets
142
Roys Lattice Space
143
Cantors Leaky Tent
145
A PseudoArc
147
Millers Biconnected Set
148
Wheel without Its Hub
150
Bounded Metrics
151
Sierpinskis Metric Space
152
Duncans Space
153
Cauchy Completion
154
The Post Office Metric
155
Radial Interval Topology
156
Bings Discrete Extension Space
157
METRIZATION THEORY
160
Conjectures and Counterexamples
161
APPENDICES
184
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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