Nowadays most students of mathematics are acquainted with the language of categories and functors. In this book we shall be dealing not only with the basic category of topological spaces and continuous functions but also with various other categories associated with it. Consequently we begin with a preliminary chapter in which some of the common features of these categories are discussed in general terms. This avoids a certain amount of repetition and enables the essential points to be made more clearly. Illustrations are taken from the category of sets, which underlies all the other categories we shall be considering.