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.

This article develops an interdisciplinary framework that applies topological and graph-theoretical methods to public procurement markets and digital platform economies. Conceptualizing legal–economic interactions as dynamic networks of nodes and edges, we show how structural properties—centrality, clustering, connectivity, and boundary formation—shape contestability, resilience, and compliance. Using EU-relevant contexts (public procurement directives and the Digital Markets Act), we formalize network representations for buyers, suppliers, platforms, and regulators; define operational indicators; and illustrate an empirical, value-weighted buyer → supplier network to reveal a sparse but highly modular architecture with a high-value backbone. We then map these structural signatures to concrete legal levers (lotting and framework design, modification scrutiny, interoperability and data-access duties) and propose dashboard-style diagnostics for proactive oversight. The findings demonstrate how topological modelling complements doctrinal analysis by making hidden architectures visible and by linking measurable structure to regulatory outcomes. We conclude with implications for evidence-informed regulatory design and a research agenda integrating graph analytics, comparative evaluation across jurisdictions, and machine-learning-assisted anomaly detection.