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Abstract
The perm of the theory of proximity spaces showed itself as early as 1908 [78]. The subject was essentially rediscovered in the early 1950’s by Efremovic [28,29| when he axiomatically characterized the proximity relation ’A is near B’ for subsets A and B of any set X. The set X together with this relation was called an infinitesimal (proximity) space, and is a natural generalization of a metric space and of a topological group.
Efremovic later used proximity neighbourhoods to obtain an equivalent set of axioms for a proximity space and thereby an alternative approach to the theory.
Defining the closure of a subset A of X to be the collection
of all points of X ’near’ A, Efremovic [29] showed that a topology
can be introduced in a proximity space and that one thereby
obtains completely regular (and hence uniformizable) space. He
further showed that every completely regular space X can be
turned into a proximity space with the help of Urysohn’s function:
namely, delta(A, B) = 0 iff there exists a continuous function f mapping
X into [0, 1] such that f(A) = 0 and f(A) = 1 .
Smirnov [82] subsequently proved that every completely regular space has a maximal associated proximity space, and that it has a minimal associated proximity space if and only if it is locally compact.