Compactness is related to a number of fundamental concepts of
mathematics. Particularly important are compact Hausdorff spaces or compacta.
Compactness appeared in mathematics for the first time as one of the main
topological properties of an interval, a square, a sphere and any closed, bounded
subset of a finite dimensional Euclidean space. Once it was realized that
precisely this property was responsible for a series of fundamental facts related
to those sets such as boundedness and uniform continuity of continuous
functions 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)).