The Topological Theory of Frechet Surfaces
J. W. T. Youngs · Annals of Mathematics · 1944
The object of this paper is the study of surfaces. As the title indicates we do not propose to consider all the mathematical entities to which the term surface has been applied, and, in fact, our first obligation is to define precisely the objects of our attention. In this connection even the most casual reader will have noticed a certain degree of confusion attached to the word surface as it is sometimes used in the literature. In fact, as with some other concepts, it is a comparatively recent departure to offer a definition at all. If one begins with the premise that mathematical terminology should have a firm foundation in intuition it is perhaps difficult to justify the definition we shall offer, since a Frechet surface is ultimately defined as a certain class of mappings, and is not, strictly speaking, a geometrical object. Why then preserve the word surface in speaking of the classes we shall study? The reason is principally historical. On the other hand, though it is quite true that the layman would hardly recognize the objects of our attention as surfaces, we propose to show that the definition is a natural consequence of a desire for rigor in the intuitive approach.