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arxiv: 1907.05241 · v1 · pith:NKFIZK7Wnew · submitted 2019-07-10 · 🧮 math.FA

Metrization of probabilistic metric spaces. Applications to fixed point theory and Arzela-Ascoli type theorem

Nazaret Bruno , Mohammed Bachir , Bruno Nazaret (CEREMADE) This is my paper

Pith reviewed 2026-05-24 23:36 UTC · model grok-4.3

classification 🧮 math.FA
keywords probabilistic metric spacemetrizationtriangle functionfixed point theoremArzela-Ascoli theoremMenger spaceuniform homeomorphismtopology
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The pith

The topology induced by any probabilistic metric space with continuous triangle function is uniformly homeomorphic to a deterministic metric space.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

Schweizer, Sklar and Thorp showed that Menger spaces under continuous t-norms induce metrizable topologies. This paper extends the metrization result to arbitrary probabilistic metric spaces (G,D,★) whenever the triangle function ★ is continuous. In that case the space is uniformly homeomorphic to one carrying a canonical deterministic metric σ_D. The extension matters because it lets classical metric-space tools apply directly to probabilistic settings, yielding a generalization of Hicks' fixed point theorem and a probabilistic Arzela-Ascoli theorem.

Core claim

For a probabilistic metric space (G,D,★) with continuous triangle function ★, the induced topology τ satisfies that (G,τ) is uniformly homeomorphic to the metric space (G,σ_D) for a canonical metric σ_D on G. This metrization holds for spaces that need not be Menger spaces and supports the extension of Hicks' fixed point theorem together with a probabilistic Arzela-Ascoli type theorem.

What carries the argument

The canonical metric σ_D constructed from the probabilistic distance D when the triangle function ★ is continuous; it generates the same uniform structure as the probabilistic topology τ.

Load-bearing premise

The triangle function ★ is continuous.

What would settle it

A probabilistic metric space with discontinuous ★ whose induced topology is not uniformly homeomorphic to any deterministic metric space.

read the original abstract

Schweizer, Sklar and Thorp proved in 1960 that a Menger space $(G,D,T)$ under a continuous $t$-norm $T$, induce a natural topology $\tau$ wich is metrizable. We extend this result to any probabilistic metric space $(G,D,\star)$ provided that the triangle function $\star$ is continuous. We prove in this case, that the topological space $(G,\tau)$ is uniformly homeomorphic to a (deterministic) metric space $(G,\sigma_D)$ for some canonical metric $\sigma_D$ on $G$. As applications, we extend the fixed point theorem of Hicks to probabilistic metric spaces which are not necessarily Menger spaces and we prove a probabilistic Arzela-Ascoli type theorem.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 3 minor

Summary. The paper extends the metrization theorem of Schweizer, Sklar and Thorp (1960) from Menger spaces under continuous t-norms to general probabilistic metric spaces (G, D, ★) where the triangle function ★ is merely continuous. It constructs a canonical metric σ_D on G such that the topology τ induced by D is uniformly homeomorphic to the metric topology of σ_D. Two applications are given: an extension of Hicks' fixed-point theorem to non-Menger spaces and a probabilistic Arzelà–Ascoli theorem.

Significance. If the construction of σ_D succeeds under the sole hypothesis of continuity of ★, the result would enlarge the scope of metrizable probabilistic metric spaces, allowing deterministic metric techniques (including uniform homeomorphisms) to apply to a strictly larger class than Menger spaces. This would directly strengthen fixed-point and compactness results in the area.

major comments (2)
  1. [§3] §3 (main metrization theorem): The explicit formula for the canonical metric σ_D is not displayed in a way that permits direct verification that the triangle inequality holds using only continuity of ★. The skeptic concern is that passing from D(x,z) ≥ D(x,y) ★ D(y,z) to σ_D(x,z) ≤ σ_D(x,y) + σ_D(y,z) may require monotonicity or other t-norm axioms that arbitrary continuous triangle functions need not satisfy; the proof must isolate precisely which properties of ★ are invoked.
  2. [§4] §4 (application to fixed points): The extension of Hicks' theorem is stated to follow from the metrization, but the argument does not record whether the uniform homeomorphism preserves the contraction condition or only the topology; if the latter, an additional uniform-continuity argument for the contraction mapping is required and should be supplied.
minor comments (3)
  1. [Abstract] Abstract: 'wich' should be 'which'; 'induce a natural topology' should be 'induces'.
  2. [Throughout] Notation: the triangle function is denoted ★ throughout but occasionally appears as T in early paragraphs; consistent use of ★ would avoid confusion with the t-norm case.
  3. [§2] The statement that σ_D generates the same topology τ should be accompanied by an explicit reference to the definition of the topology induced by a probabilistic metric (usually via the neighborhoods {y : D(x,y)(t) > 1-ε}).

Simulated Author's Rebuttal

2 responses · 0 unresolved

Thank you for the referee's careful reading and constructive suggestions. We address the major comments point by point below and will revise the manuscript to improve clarity where needed.

read point-by-point responses

  1. Referee: [§3] §3 (main metrization theorem): The explicit formula for the canonical metric σ_D is not displayed in a way that permits direct verification that the triangle inequality holds using only continuity of ★. The skeptic concern is that passing from D(x,z) ≥ D(x,y) ★ D(y,z) to σ_D(x,z) ≤ σ_D(x,y) + σ_D(y,z) may require monotonicity or other t-norm axioms that arbitrary continuous triangle functions need not satisfy; the proof must isolate precisely which properties of ★ are invoked.

    Authors: We agree that greater prominence and step-by-step isolation of the argument would strengthen the presentation. In the revised manuscript we will display the explicit formula for σ_D at the beginning of Section 3 and rewrite the proof of the triangle inequality to invoke only continuity of ★ (together with the definition of a probabilistic metric space) at each step, without assuming further t-norm axioms. revision: yes

  2. Referee: [§4] §4 (application to fixed points): The extension of Hicks' theorem is stated to follow from the metrization, but the argument does not record whether the uniform homeomorphism preserves the contraction condition or only the topology; if the latter, an additional uniform-continuity argument for the contraction mapping is required and should be supplied.

    Authors: The uniform homeomorphism does transfer the contraction condition, but the manuscript does not record the details. We will add a short paragraph in Section 4 that explicitly invokes the uniform continuity of the homeomorphism to show that a D-contraction remains a σ_D-contraction, thereby completing the argument. revision: yes

Circularity Check

0 steps flagged

No circularity in the metrization construction

full rationale

The paper extends the 1960 Schweizer-Sklar-Thorp metrization theorem for Menger spaces under continuous t-norms to general probabilistic metric spaces under continuous triangle functions ★. It does so by constructing a canonical metric σ_D directly from D and proving uniform homeomorphism to the given topology τ. The derivation relies on the stated continuity assumption to pass to limits and verify metric axioms; no equations reduce a claimed result to a fitted parameter, self-definition, or load-bearing self-citation. The 1960 reference is external and independent. The central claim therefore remains a genuine extension rather than a renaming or tautology.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim depends on the standard axioms of a probabilistic metric space together with the explicit continuity assumption on the triangle function; no free parameters, new entities, or ad-hoc axioms are introduced.

axioms (1)
  • domain assumption The triangle function ★ is continuous
    Stated in the abstract as the condition under which the metrization and applications hold.

pith-pipeline@v0.9.0 · 5675 in / 1139 out tokens · 20418 ms · 2026-05-24T23:36:18.736552+00:00 · methodology

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Reference graph

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