A generalisation of the functional approach to compactness

A topological space X is compact iff the projection π : X × Y → Y is closed for any space Y. Taking this as a definition and then asking that π maps α-closed subspaces of X × Y onto β-closed subspaces of Y, for different closures α and β, extends the notion of compactness to include also examples of “asymmetric compactness” pursued in the article. rical closure operators and a so-called “functional approach to general topology” are employed to define and prove fundamental properties of compact objects and proper maps in this generalised setting.