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

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Document pages: 11 pages

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 thatguarantees the completeness and transitivity of its soft part, and a (ii)characterization of a connected topological space in terms of its attendantproperties on the space. Our work generalizes antecedent results in appliedmathematics, all following Eilenberg (1941), and now framed in the context of aparametrized-topological space. This re-framing is directly inspired by thecontinuity assumption in Wold (1943-44) and the mixture-space structureproposed in Herstein and Milnor (1953), and the unifying synthesis of thesepioneering but neglected papers that it affords may have independent interest.

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