On fuzzy pseudo-normed vector spaces

Abstract

We provide a method for introducing fuzzy pseudo-metric topologies on sets, and fuzzy pseudo-normed topologies on vector spaces over R or C which will be fuzzy linear topologies. We define fuzzy pseudo-metrics for pairs of crisp points, and fuzzy pseudo-norms for crisp points, as fuzzy real numbers ⩾ 0 (as defined by Hutton). We define the associated fuzzy open balls, with crisp points for their centres and fuzzy real numbers > 0 for their radii. These form a basis for the associated fuzzy pseudo-metric topology. The axioms governing fuzzy pseudo-metric and fuzzy pseudo-norm are straightforward extensions for the corresponding axioms in the crisp case. The formulation conforms with Zadeh's Extension Principle. We show that Katsaras' concept of fuzzy seminorm (Fuzzy Sets and Systems 12 (1984) 143–154) is equivalent to ours, in the sense that both concepts will result in same fuzzy linear topologies.