Semifield metric spaces

Abstract

Extending the results of An†onovskiĭ, Bol†janskiĭ, and Sarymsakov on semifield metric spaces, the authors define a regular semifield metric to be one in which the distance in the standard Tychonoff product representation of a point from a disjoint closed set is nonzero. It is shown that every completely regular topological space possesses a completely regular semifield metric and that there is an equivalent completely regular semifield metric for every semifield metric space. A normal semifield metric is defined to be one in which the distance between two disjoint closed sets is nonzero and it is shown that possessing a normal semifield metric is equivalent to being a normal topological space. Finally, Cauchy nets in semifield metric spaces are introduced leading to the notion of completeness. It is shown that a semifield metric space is complete iff every Cauchy net with the property that its directed set has cardinality less than or equal to the cardinality of the indexing set of the Tychonoff product representation of the semifield converges.