Compactifications of products of metric spaces and their relations to Čech-Stone and Smirnov compactifications

AbstractThis paper is concerned with compactness and compactifications in the epireflective hull of the subcategory of extended pseudometric spaces of the category of approach spaces. Since in this context, compactness is epireflective, this provides a first characterization of a compactification which is the equivalent of the Čech-Stone compactification for topological spaces. An alternative construction of this compactification reveals that its topological coreflection is the Smirnov compactification of a canonically associated proximity relation. An interesting result which ensues is the extension of the Euclidean metric on the natural numbers to a distance on the underlying set of the Čech-Stone compactification which yields the Čech-Stone compactification as topological coreflection