International Journal of Topology

The field of probabilistic metric spaces has an intrinsic interest based on a blend of ideas drawn from metric space theory and probability theory. The goal of the present paper is to introduce and study new ideas in this field. In general terms, we investigate the following concepts: linearly ordered families of distances and associated continuity properties, geometric properties of distances, finite range weak probabilistic metric spaces, generalized Menger spaces, and a categorical framework for weak probabilistic metric spaces. Hopefully, the results will contribute to the foundations of the subject.

We prove that in every nonempty perfect Polish space, every dense $G_{\delta }$ subset contains strictly decreasing and strictly increasing chains of dense $G_{\delta }$ subsets of length $\mathfrak{c}$, the cardinality of the continuum. As a corollary, this holds in ${\mathbb{R}}^n$ for each $n\ge 1$. This provides an easy answer to a question of Erdős since the set of Liouville numbers admits a descending chain of cardinality $\mathfrak{c}$, each member of which has the Erdős property. We also present counterexamples demonstrating that the result fails if either the perfection or the Polishness assumption is omitted. Finally, we show that the set $\mathcal{T}$ of real Mahler T-numbers is a dense Borel set and contains a strictly descending chain of length $\mathfrak{c}$ of proper dense Borel subsets.

We propose a geometry topological framework to analyze storm dynamics by coupling persistent homology with Anti-de Sitter (AdS)-inspired metrics. On radar images of a bow echo event, we compare Euclidean distance with three compressive AdS metrics ($\alpha$ = 0.01, 0.1, 0.3) via time-resolved $H_1$ persistence diagrams for the arc and its internal cells. The moderate curvature setting ($\alpha =0.1$) offers the best trade-off: it suppresses spurious cycles, preserves salient features, and stabilizes lifetime distributions. Consistently, the arc exhibits longer, more dispersed cycles (large-scale organizer), while cells show shorter, localized patterns (confined convection). Cross-correlations of $H_1$ lifetimes reveal a temporal asymmetry: arc activation precedes cell activation. A differential indicator $\Delta \left(t\right)$ based on Wasserstein distances quantifies this divergence and aligns with the visual onset in radar, suggesting early warning potential. Results are demonstrated on a rapid Corsica bow echo; broader validation remains future work.