Vol. 37, No. 2, 1971

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Proximity bases and subbases

Prem Lal Sharma

Vol. 37 (1971), No. 2, 515–526
Abstract

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.

Mathematical Subject Classification 2000
Primary: 54E05
Milestones
Received: 30 November 1970
Published: 1 May 1971
Authors
Prem Lal Sharma