The Sierpinski-Erdös Duality Theorem

Abstract

The principal use for the notion of category is in the formulation of existence proofs. This may be termed its telescopic function. Baire’s theorem enables us to bring into focus mathematical objects which otherwise may be difficult to see! But the study of category serves another purpose, too. By developing the theories of measure and category simultaneously, and by calling attention to their points of similarity and difference, we have tried to show how the two theories illuminate each other. Because the theory of measure is more extensive and “important” than that of category, the service is mainly in the direction of measure theory. This may be termed the stereoscopic function of the study of category; it adds perspective to measure theory! The suggestion to look for a category analogue, or a measure analogue, has very often proved to be a useful guide. In this and the following chapters we shall take a closer look at the duality we have observed between measure and category, to see how far it extends in the case of the line and other spaces, and to discover what underlies it.