Statistical metric spaces as related to topological spaces

Abstract

Our discussion answers the questions as to what topological spaces are statistically metrizable in the sense of Schweizer and Sklar [5] and whether this can be discerned by the t-norm on the space in the Menger triangle relation. Namely we prove (1) the class of topological Menger spaces coincides with that of semi-metrizable topological spaces, and (2) no condition weaker than 1=sup_x<1T(x, x) can guarantee that a Menger space satisfying the Menger triangle relation under T is topological.