Hausdorff Compactifications: A Retrospective

Abstract

It was far from clear in the early years of general topology how to abstract the proper concepts from ℝ or I, first to metric spaces and later, to topological spaces. Today it seems that topology of the early 20th century was a virtual battleground of competing ideas, with various notions vying for attention. To quote Engelking [1989], pp. 132–133 on the topic of compactness: When general topology was in its infancy, defining new classes of spaces often consisted in taking a property of the closed interval I or of the real line ℝ and considering the class of all spaces that have this property; classes of separable, compact, complete and connected spaces were defined following this pattern. At first this method was used to define some classes of metric spaces, later definitions were extended to topological spaces. Sometimes properties equivalent in the class of metric spaces, when extended to topological spaces, led to different classes of topological spaces ... and it was not immediately clear which class was the proper generalization. This happened with compactness, and for some time there was doubt whether the proper extension of the class of compact metric spaces is the class of compact spaces, the class of countable compact spaces, or the class of sequentially compact spaces ... By now, it is quite clear that it is the class of compact spaces; this class behaves best with respect to operations on topological spaces, is most often met in applications and leads to the most interesting problems.