In this paper we provide a
definition of a proximity-base (subbase); this enables us to prove results
analogous to those in topological and uniform spaces. For example we prove
that the set of all proximities on a set X forms a complete lattice. Another
consequence is that a proximity on a set X can be defined as a certain collection of
pseudomelrics on X. A pseudometric approach to proximities is discussed
in [4]. Two definitions of a “proximity base” have been given in literature,
one by Császár and Mrowka [1] and the other by Njasted [3]. Neither
of these definitions is perfectly satisfactory; the first does not determine a
unique proximity whereas for the second (i) it is not known whether every
proximity has such a base and (ii) a proximity itself is not a base unless it is
discrete.
