The creative process of mathematics, both historically and individually, may be described as a counterpoint between theorems and examples. Alยญ though it would be hazardous to claim that the creation of significant examples is less demanding than the development of theory, we have disยญ 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 necesยญ 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 underยญ 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 topologiยญ 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, theoยญ 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
CONTENT
I Basic Definitions -- 1. General Introduction -- 2. Separation Axioms -- 3. Compactness -- 4. Connectedness -- 5. Metric Spaces -- II Counterexamples -- III Metrization Theory -- Conjectures and Counterexamples -- IV Appendices -- Special Reference Charts -- Separation Axiom Chart -- Compactness Chart -- Paracompactness Chart -- Connectedness Chart -- Disconnectedness Chart -- Metrizability Chart -- General Reference Chart -- Problems -- Notes
