General Topology II : Compactness

Compactness is related to a number of fundamental concepts of mathemat- ics. Particularly important are compact Hausdorff spaces or compacta. Com- pactness appeared in mathematics for the first time as one of the main topo- logical properties of an interval, a square, a sphere and any closed, bounded subset of a finite dimensional Euclidean space. Once it was realized that pre- cisely this property was responsible for a series of fundamental facts related to those sets such as boundedness and uniform continuity of continuous func- tions defined on them, compactness was given an abstract definition in the language of general topology reaching far beyond the class of metric spaces. This immensely extended the realm of application of this concept (including in particular, function spaces of quite general nature). The fact, that general topology provided an adequate language for a description of the concept of compactness and secured a natural medium for its harmonious development is a major credit to this area of mathematics. The final formulation of a general definition of compactness and the creation of the foundations of the theory of compact topological spaces are due to P.S. Aleksandrov and Urysohn (see Aleksandrov and Urysohn (1971)).