Counterexamples in TopologyThe 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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arc connected basis element basis neighborhood 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 Hausdorff space homeomorphic hyperconnected integers interval topology limit point Lindelöf locally compact locally connected locally finite 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 pseudocompact quasicomponent rational real line regular spaces screenable second category second countable semimetric separation axioms sequence sequentially compact Show space is completely space is metrizable subspace T₁ space 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