Counterexamples in TopologySpringer Science & Business Media, 6 Ara 2012 - 244 sayfa 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. |
İçindekiler
3 | |
Global Compactness Properties | 18 |
Metric Spaces | 34 |
Complete Metric Spaces | 36 |
Metrizability | 37 |
Metric Uniformities | 38 |
COUNTEREXAMPLES | 39 |
Finite Discrete Topology | 41 |
Michaels Product Topology | 105 |
Tychonoff Plank | 106 |
Alexandroff Plank | 107 |
Dieudonne Plank | 108 |
Tychonoff Corkscrew | 109 |
Hewitts Condensed Corkscrew | 111 |
Thomas Plank 113 94 Thomas Corkscrew | 113 |
Weak Parallel Line Topology | 114 |
Indiscrete Topology | 42 |
Partition Topology | 43 |
Finite Particular Point Topology | 44 |
Finite Excluded Point Topology | 47 |
EitherOr Topology | 48 |
Finite Complement Topology on a Countable Space | 49 |
Countable Complement Topology | 50 |
Compact Complement Topology | 51 |
Countable Fort Space | 52 |
Fortissimo Space | 53 |
ArensFort Space | 54 |
Modified Fort Space | 55 |
Euclidean Topology | 56 |
The Cantor Set | 57 |
The Rational Numbers | 59 |
Special Subsets of the Real Line | 60 |
Special Subsets of the Plane | 61 |
One Point Compactification Topology | 63 |
Hilbert Space | 64 |
Hilbert Cube | 65 |
Order Topology | 66 |
Open Ordinal Space 0г г | 68 |
Uncountable Discrete 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 71 Discrete Irrational Extension of R | 90 |
Double Origin 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 |
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 Rw | 124 |
0N I¹ | 126 |
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 150 133 Tangoras Connected Space | 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 |
228 | |
236 | |
Diğer baskılar - Tümünü görüntüle
Sık kullanılan terimler ve kelime öbekleri
arc connected basis element basis neighborhood C₁ clearly closed set closed subset closure collectionwise normal completely normal completely regular conjecture converges countable local basis countably paracompact counterexamples define dense subset dense-in-itself discrete topology disjoint open sets equivalent Euclidean topology extremally disconnected finite subcover fully normal function f Hausdorff space homeomorphic hyperconnected integers interval topology limit point Lindelöf locally compact locally connected metacompact Moore space nonempty normal Moore space normal space o-compact open cover open interval open neighborhoods open set containing order topology ordinal space path connected point topology properties pseudocompact quasicomponent real line regular spaces screenable second category second countable semimetric separation axioms sequence sequentially compact Show space is completely space is metrizable subspace T₁ T3 space theorem topological space topology Example totally disconnected totally separated Tychonoff ultraconnected ultrafilters uncountable union unit interval Urysohn w-accumulation point weakly countably compact zero dimensional