Eraldo Giuli; Maria Manuel Clementino; Walter Tholen;
Categorical Foundations. Special Topics in Order, Topology, Algebra, and Sheaf Theory.
Chapter 3: A Functional Approach to General Topology
In this chapter we wish to present a categorical approach to fundamental concepts of General Topology, by providing a category X with an additional structure which allows us to display more directly the geometric properties of the objects of X regarded as spaces. Hence, we study topological properties for them, such as Hausdorff separation, compactness and local compactness, and we describe important topological constructions, such as the compact-open topology for function spaces and the Stone-Cech compactification. Of course, in a categorical setting, spaces are not investigated "directly" in terms of their points and neighbourhoods, as in the traditional set-theoretic setting; rather, one exploits the fact that the relations of points and parts inside a space become categorically special cases of the relation of the space to other objects in its category. It turns out that many stability properties and constructions are established more economically in the categorical rather than the set-theoretic setting, leave alone the much greater level of generality and applicability. (...)
Maria Cristina Pedicchio and Walter Tholen (ed.)
Encyclopedia of Mathematics and its Applications 97, Cambridge University Press