On an Extension of a Theorem of Eilenberg and a Characterization of Topological Connectedness

TL;DR

This paper generalizes Eilenberg's theorem by introducing a p-continuity assumption for bi-relations, providing a new characterization of topological connectedness within a parametrized-topological space framework.

Contribution

It extends classical results by integrating semi-transitive bi-relations with topological properties, unifying earlier theories in a broader, more general setting.

Findings

p-continuity guarantees completeness and transitivity of the soft relation

Characterization of connected topological spaces via bi-relations

Generalization of Eilenberg's theorem in a parametrized-topological context

Abstract

On taking a non-trivial and semi-transitive bi-relation constituted by two (hard and soft) binary relations, we report a (i) p-continuity assumption that guarantees the completeness and transitivity of its soft part, and a (ii) characterization of a connected topological space in terms of its attendant properties on the space. Our work generalizes antecedent results in applied mathematics, all following Eilenberg (1941), and now framed in the context of a parametrized-topological space. This re-framing is directly inspired by the continuity assumption in Wold (1943-44) and the mixture-space structure proposed in Herstein and Milnor (1953), and the unifying synthesis of these pioneering but neglected papers that it affords may have independent interest.