Probabilistic norms on the homeomorphisms of a group

Abstract

In this paper, a probabilistic metric on the set of all auto-homeomorphisms of a group is presented. We show that the defined probabilistic group metric is right invariant and it implies a probabilistic group norm. In addition, by the probabilistic norm admissibility condition, we study the uniform continuity of homeomorphisms, and finally, we prove some theorems about topologically equivalent probabilistic norms.