Metrization of proximity spaces
Solomon Leader · Proceedings of the American Mathematical Society · 1967
0. Introduction. Our purpose here is to present a metrization criterion for proximity spaces that is analogous to R. L. Moore's criterion for topological spaces [8]. Where Moore's criterion demands a sequence of open coverings, our criterion requires admissible coverings. In Moore's criterion a set A is close to a point b (that is, b A) if and only if the star of A meets the star of b for every covering in the sequence. For proximity spaces we require only that a set A be close to a set B if and only if the star of A meets B for every covering in the sequence. Such weakening of the star-separation condition is possible because proximity spaces have a strong separation axiom. However, we must require that our sequence of coverings be nested by refinement. Our covering criterion is easily translated into a criterion involving admissible entourages. These entourages play a key role in the criterion of Efremovic and Svarc [6]. Their criterion easily implies ours, but the converse is much more difficult. It should therefore be easier to prove a given proximity space metrizable using our criterion rather than that of [6].