This book has two main themes： the Baire cate8ory theorem as a method for proving existence， and the /"duality/" between measure and category. The category method is illustrated by a variety of typical applications， and the analogy between measure and category is explored in all of its ramifications. To this end， the elements of metric topology are reviewed and the principal properties of Lebesgue measure are derived. It turns out that Lebesgue integration is not essential for present purposes-the Riemann integral is sufficient. Concepts of general measure theory and topology are introduced， but not just for the sake of 8cncrality， Needless to say， the term /"category/" refers always to Bairc category； it has nothing to do with the term as it is used in homological algebra.
　　The book is a revised and expanded version of notes originally prepared for a course of lectures given at Haverford College during the spring of 1957 under the auspices of the William Pyle Philips Fund. These， in turn， were based on the Earle Raymond Hedrick Lectures presented at the Summer Meeting of the Mathematical Association of America at Seattle， Washington， in August， 1956.