L.A. Steen - Böcker

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.

Mathematics today is approaching a state of cnSIS. As the demands of science and society for mathematical literacy increase, the percentage of American college students intending to major in mathematics plummets and achievement scores of entering college students continue thelt unremit­ ting decline. As research in core mathematics reaches unprecedented heights of power and sophistication, the growth of diverse applied special­ ties threatens to fragment mathematics into distinct and frequently hostile mathematical sciences. These crises in mathematics presage difficulties for science and engineer­ ing, and alarms are beginning to sound in the scientific and even in the political communities. Citing a trend towards "virtual scientific and techno­ logical illiteracy" and a "shrinking of our national commitment to excel­ lence . . . in science, mathematics and technology," a recent study con­ ducted for the President by the U. S. National Science Foundation and Department of Education warns of serious impending shortcomings in public understanding of science. "Today people in a wide range of non­ scientific . . . professions must have a greater understanding of technology than at any time in our history. Yet our educational system does not now provide such understanding. " The study goes on to conclude that present trends pose great risk of manpower shortages in the mathematical and engineering sciences. "The pool from which our future scientific and engineering personnel can be drawn is . . . in danger of becoming smaller, even as the need for such personnel is increasing. " It is time to take a serious look at mathematics tomorrow.