Translation-invariant generalized topologies induced by probabilistic norms

Abstract

One considers probabilistic normed spaces as defined by Alsina,Schweizer and Sklar, but with non necessarily continuous triangle functions. Such spaces are endowed with a generalized topology that is Fréchet-separated, translation-invariant and countably generated by radial and circled 0-neighborhoods. Conversely, we show that such generalized topologies are probabilistically normable.