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-Čech compactification. Of course, in a categorical setting, spaces are not investigated “directly” in terms of their points and neighborhoods, 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.

The idea of providing a category with some kind of topological structure is certainly not new. So-called Grothendieck topologies (see Chapter VII) and, more generally, Lawvere-Tierney topologies are fundamental for the geometrically inspired construction of topoi. Specifically, these structures provide a notion of closure and thereby a notion of closed subobject, for every object in the category, such that all morphisms become “continuous”. The notion of Dikranjan-Giuli closure operator [17] axiomatizes this idea and can be used to study topological properties categorically (see, for example, [9, 12]).