[Paper Review] Translation-invariant generalized topologies induced by probabilistic norms

Bernardo Lafuerza–Guillén, Jose L. Rodrıguez-Blancas|arXiv (Cornell University)|May 23, 2005

Fuzzy and Soft Set Theory8 references3 citations

TL;DR

This paper establishes a characterization of translation-invariant generalized topologies on real vector spaces that arise from Menger probabilistic normed spaces (PN spaces) with non-continuous triangle functions. It proves that such a topology is derivable from a PN space if and only if the 0-neighborhood filter at the origin admits a countable base of radial and circled sets, under mild conditions on the t-norm T, extending probabilistic metrization theory to non-continuous settings.

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 Fréchet-separable, translation-invariant and countably generated by radial and circled 0-neighborhoods. Conversely, we show that such generalized topologies are probabilistically normable.

Motivation & Objective

• To extend the theory of probabilistic normed spaces beyond continuous triangle functions.
• To characterize generalized topologies induced by probabilistic norms in vector spaces.
• To establish conditions under which such generalized topologies are probabilistically normable.
• To generalize Höhle’s probabilistic metrization result to the setting of probabilistic norms.
• To explore the relationship between probabilistic normability and boundedness in PN spaces with non-continuous structures.

Proposed method

• Uses a t-norm T satisfying sup₀≤x<1 T(x,x) < 1 and T(x,y) ≤ xy for small x,y to define triangle functions τ_T and τ_T*.
• Constructs a probabilistic norm ν on a vector space S using a countable filter base {V_n} of radial and circled 0-neighborhoods.
• Defines ν_p(x) via distribution functions F_n that depend on the index n of the neighborhood V_n.
• Applies the generalized topology framework from Fréchet and Höhle to ensure translation-invariance and Fréchet-separation.
• Verifies that (S, ν, τ_T, τ_T*) satisfies all axioms of a Menger PN space, including the Šerstnev condition.
• Demonstrates that the generalized topology induced by ν matches the original topology via equivalence of filter bases.

Experimental results

Research questions

• RQ1 Under what conditions is a translation-invariant, Fréchet-separated generalized topology on a real vector space derivable from a Menger PN space with non-continuous triangle functions?
• RQ2 What topological properties (e.g., radiality, circledness) of the 0-neighborhood filter are necessary and sufficient for probabilistic normability?
• RQ3 How do t-norms with local submultiplicativity (T(x,y) ≤ xy near 0) affect the structure of probabilistic norms?
• RQ4 Can the probabilistic metrization result of Höhle be extended from PM spaces to PN spaces under weaker continuity assumptions?
• RQ5 What is the relationship between ν-induced generalized topologies and D-boundedness in PN spaces with non-continuous triangle functions?

Key findings

• A Fréchet-separated, translation-invariant generalized topology on a real vector space S is derivable from a Menger PN space (S, ν, τ_T, τ_T*) if and only if the 0-neighborhood filter admits a countable base of radial and circled sets.
• The t-norm T must satisfy sup₀≤x<1 T(x,x) < 1 and T(x,y) ≤ xy for small x,y (for some δ > 0), ensuring local submultiplicativity.
• The probabilistic norm ν is explicitly constructed from a countable filter base {V_n} of radial and circled 0-neighborhoods via distribution functions F_n with decreasing values.
• Axiom (N3) — the probabilistic triangle inequality — holds due to the bound τ_T(ν_p, ν_q)(x) ≤ 1 - 1/N₀ ≤ ν_{p+q}(x), where N₀ is chosen so 1 - 1/N₀ ≥ sup T(x,x).
• Axiom (N4) — the convexity condition — is verified using the t-conorm τ_T* and the submultiplicativity of T, ensuring ν_p ≤ τ_T*(ν_λp, ν_(1−λ)p).
• The generalized topology induced by ν is equivalent to the original topology, as the filter base {p : ν_p(1/(n+1)) ≥ 1 - 1/(N₀(n+1))} matches the original {V_n}.