Proximities on Tychonoff spaces : Smirnov theorem

Abstract

In the following text we define the axioms of a proximity relation δ as defined by Efremovic and explore some properties derived from these basic rules. We briefly discuss two alternative approaches to defining a proximity relation on a set X, first by the dual binary relation ≪ called a proximity neighbourhood relation, and later by the use of a collection of clusters σ ∈ P (P (X)). We recall the topological separation axioms up to T4, and define a relation between proximities and topologies by way of “compatibility”. We further explore proximal mappings and equimorphisms as parallel structures to continuous mappings and homeomorphisms, and their associated properties. The properties of clusters are explored with consideration for how they mirror ultrafilters, and we expand upon this relation to prove many of their properties. Next, we introduce Hausdorff compactifications, i.e. T2 compact extensions, of a Tychonoff topological space and prove some very important results in this area. From there we follow in the footsteps of Smirnov to establish a one-to-one correspondence between the set of all compatible proximities and the set of all T2 compactifications on a Tychonoff space X.