ULTRAFILTERS AND COMPACTIFICATION OF UNIFORM SPACES í1)

In point set topology—or, more precisely, in the part of it once called "theory of abstract spaces"—there are two principal methods of investigation. The first one refers to the topological space alone(2) ; we shall call it the internal method : for example the separation axioms Po, Pi, T2 [A-H], the notions of regularity, normality, compactness, are expressed in terms of the topological space only. The second method uses the real numbers as a tool for investigating the topological space S, and will be called the external method. Here the real numbers appear via the channel of real valued continuous functions defined on 5. One may consider either a restricted class of these functions, like distances, pseudo distances, separating functions, or the whole ring of all continuous real valued functions defined on S. Examples of this method are the notions of complete regular space and of metric space, the Tietze extension theorem, the reconstruction of the space from its ring of real valued continuous functions [G](3). In both methods one recognizes rapidly that, in order for the topological space to have many interesting properties, one must impose more restrictive conditions. About these conditions, mathematicians now seem in better agreement than some years ago [W] [Tu]. From the internal standpoint the interesting spaces are the uniform spaces(4) and the compact spaces(6). From the external standpoint, if one wants the topological space described accurately