On an extension of a theorem of Eilenberg and a characterization of topological connectedness

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 [14], and now framed in the context of a parametrized-topological space. This re-framing is directly inspired by the continuity assumption in Wold [50] and the mixture-space structure proposed in Herstein and Milnor [27], and the unifying synthesis of these pioneering but neglected papers that it affords may have independent interest.