On Quasi-Modular Pseudometric Spaces and Asymmetric Uniformities

We study quasi-modular pseudometric spaces as asymmetric refinements of modular metric structures. To each such space we associate canonical forward and backward quasi-uniformities and the corresponding directional topologies. We introduce directional notions of convergence, completeness, total boundedness, and compactness, and show that these properties are not preserved under symmetrization. In particular, forward and backward completeness may differ, and compactness of the symmetrized uniformity does not imply directional compactness. Using enriched category theory as a comparison framework, we show that symmetrization yields a symmetric enriched category whose Cauchy completion coincides with the classical uniform completion, while directional notions remain invisible at this level.

💡 Research Summary

The paper investigates a new class of spaces called quasi‑modular pseudometric spaces, which arise by dropping the symmetry requirement from the classical modular metric framework while retaining the scale‑dependent triangle inequality and monotonicity in the scale parameter. A quasi‑modular pseudometric is a family {wλ}λ>0 of non‑negative functions on X×X satisfying three axioms: (QM1) wλ(x,x)=0 for all x and λ, (QM2) the modular triangle inequality wλ+µ(x,z) ≤ wλ(x,y)+wµ(y,z) for all points and all positive scales, and (QM3) the map λ↦wλ(x,y) is right‑continuous and non‑increasing. This setting generalises the symmetric modular distances introduced by Chistyakov and includes, as special cases, scaled ordinary metrics (e.g., wλ(x,y)=d(x,y)/λ) and Luxemburg‑type gauges derived from Orlicz spaces.

From a given quasi‑modular pseudometric w the authors construct two canonical quasi‑uniformities: the forward quasi‑uniformity V⁺ generated by basic entourages V⁺(λ,ε) = {(x,y) | wλ(x,y) < ε} and the backward quasi‑uniformity V⁻ generated by V⁻(λ,ε) = {(x,y) | wλ(y,x) < ε}. These are genuine quasi‑uniformities because the t‑conorm ⊕ (often taken as max) guarantees the small‑compose property: for any entourage there exist smaller parameters r′<r and t′,t″>0 such that the composition of two r′‑entourages is contained in the original r‑entourage. The forward and backward quasi‑uniformities give rise to two distinct topologies τ⁺ and τ⁻, defined by forward and backward neighbourhood bases B⁺(x;r,t) = {y | w_t(x,y) < r} and B⁻(x;r,t) = {y | w_t(y,x) < r}. The pair (τ⁺,τ⁻) forms a canonical bitopological space associated with the quasi‑modular gauge.

A symmetrisation procedure is introduced by defining a symmetric gauge w_sym(λ)(x,y) = max{wλ(x,y), wλ(y,x)} (or more generally wλ(x,y)⊕wλ(y,x)). The uniformity generated by the symmetric entourages E_sym(r,t) = V⁺(r,t) ∩ V⁻(r,t) coincides with the join topology τ⁺ ∨ τ⁻. When w is already symmetric, the forward and backward structures collapse to the usual uniform space.

The authors then develop directional notions of convergence, Cauchy sequences, precompactness and compactness. A sequence (x_n) forward‑converges to x if for every λ,ε there exists N such that wλ(x_n,x) < ε for all n≥N; backward convergence swaps the arguments. Analogously, a forward Cauchy sequence requires that for every λ,ε there is N with wλ(x_n,x_m) < ε for all n,m≥N, while backward Cauchy uses wλ(x_m,x_n). The paper proves that forward completeness (every forward Cauchy sequence forward‑converges) and backward completeness are independent: examples are given where one holds but the other fails. Moreover, the symmetrised uniform space may be complete even when the original quasi‑modular space is not forward or backward complete, demonstrating that symmetrisation can hide directional completeness.

Precompactness is defined separately for the forward and backward quasi‑uniformities: a space is forward precompact if for each entourage V⁺(λ,ε) there exists a finite set F such that X ⊆ ⋃_{x∈F} V⁺