Extensions of Topological Spaces

Abstract

The undertaking of constructing spaces which contain a given space as a subspace is by no means new: the extension of the complex number plane to the complex number sphere by the addition of the one point at infinity, the extension of the real line by adjoining the two infinities ∞ and -∞, and the construction of the space of real numbers from that of the rationals by means of Cauchy sequences or Dedekind cuts are 19th Century examples of this very thing. However, only the advent of general topology made it possible to raise the general question of space extensions. It appears that the first study of problems in this area was carried out by Alexandroff and Urysohn in the early twenties [l]. Another mile stone in the history of the subject was the 1929 paper by Tychonoff in which the product theorem for compact spaces is proved and used to identify the completely regular Hausdorff spaces as precisely those spaces which can be imbedded in a compact Hausdorff space [33]. During the same period, work on certain specific extension problems was done by Freudenthal [17] and Zippin [35]. However, the first large body of systematic theory, used for the investigation of a wide range of extension problems, was presented by Stone [31] in 1937. There, one also finds the remark that "one of the interesting and difficult problems of general topology is the study of all extensions of a given space", and it appears that Stone' s own work must have convinced many others of the truth of this observation, for since that time there has been a steady succession of papers in this field. But apart from that, the study of extension spaces clearly has a very particular attraction for some mathematicians.