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.