Set Theory and Metric Spaces

I first taught a course on set theory and metric spaces in the autumn of 1949. In subsequent years Edwin Spanier presented the material in a somewhat similar way, and he prepared an excellent set of mimeographed notes. These notes were used repeatedly as a text at Chicago. I have now put them into a somewhat more definitive form. I am very grateful to Spanier for courteously allowing me to complete a project in which he was so deeply involved, and for permission to incorporate numerous exercises from his notes. The two halves of the book are of nearly equal size. The set theory (with a bow to Halmos) is super-naive. Axiomatic set theory is barely mentioned. The paradoxes get some attention, but in effect they are brushed aside as not really being menacing. My intention is to present the set theory that a working mathematician really needs ninety-nine per cent of the time, and a little bit more (on the theory that to be sure of doing enough, you must do more than enough). A little knowledge may not be a dangerous thing, but a little axiomatic set theory is just not much fun; I hold that one should either take a healthy bite or leave it out in toto. On the other hand, I hope that this book will help to fight those who say that set theory is a luxury. Hilbert vowed that no one would ever drive us out of the paradise created by Cantor. Let those who agree strive diligently to transmit to the next generation the knowledge that there is such a paradise. In the metric space half of the book I have tried to cover the basic topics with a helpful amount of detail and motivation. I hope it will be found useful by teachers who share my belief that topology is best introduced first in the less austere setting of metric spaces. A final appendix has been added to help bridge the gap between metric and topological spaces. The asterisk attached to occasional exercises probably needs no explanation. I was tempted to put a double asterisk on some. To the teacher: Assign them cautiously! I am very grateful to Rose Banfield, Fred Flowers, and Tere Shuman for their excellent job in typing drafts of the manuscript. Among the many students who made helpful comments, I wish particularly to thank Susan Bolotin Friedman and Andrew Gallant. To Judith Fiske and Carl Harris of Allyn and Bacon: my gratitude for their contribution to producing the book. IRVING KAPLANSKY