On para-uniform nearness spaces and D-complete regularity (Q1118849)

A topological space is D-completely regular if it possesses an \(F_{\sigma}\)-base. The class of D-completely regular spaces has been characterized previously by the author as the class of subspaces of products of developable spaces and contains the class of completely regular spaces. In this paper the author addresses the question of in what sense the D- completely regular spaces are ``uniformizable'' by introducing the notion of para-uniform nearness structure and characterizing the D-completely regular spaces as those for which the topology is induced by a para- uniform nearness structure. It is also shown that a topological nearness space (one for which every open cover belongs to the nearness structure) is para-uniform if and only if its induced topology is D-paracompact, a property introduced by \textit{C. M. Pareek} [Can. J. Math. 24, 1033-1042 (1972; Zbl 0265.54028)].