TOPOLOGY IN A CATEGORY: COMPACTNESS

Abstract

In a category with a subobject structure and a closure operator, we provide a categorical theory of compactness and perfectness which yields a number of classical results of general topology as special cases, including the product theorems by Tychonofi and Frol¶‡k, the existence of Stone{ • Cech compactiflcations, both for spaces and maps, and the Henriksen{Isbell characterization of perfect maps of Tychonofi spaces. Applications to other categories yield, among other things, an alternative proof for the productivity of categorically compact groups.