Translation-invariant generalized topologies induced by probabilistic norms

TL;DR

This paper explores probabilistic normed spaces with non-continuous triangle functions, showing they induce translation-invariant, Fréchet-separable generalized topologies that are probabilistically normable.

Contribution

It extends the theory of probabilistic normed spaces by characterizing the induced topologies and establishing their probabilistic normability.

Findings

Spaces are Fréchet-separable and translation-invariant

Generalized topologies are countably generated by radial and circled neighborhoods

Such topologies are shown to be probabilistically normable

Abstract

In this paper we consider probabilistic normed spaces as defined by Alsina, Sklar, and Schweizer, but equipped with non necessarily continuous triangle functions. Such spaces endow a generalized topology that is Fr\'echet-separable, translation-invariant and countably generated by radial and circled 0-neighborhoods. Conversely, we show that such generalized topologies are probabilistically normable.