Hausdorff Spaces in Bermejo Algebras: The Birth of Treonic Manifold Construction

Abstract

Alejandro Jesus Bermejo Valdes

We conducted a topological analysis of the novel, non-associative, and unital algebraic structure known as Bermejo Algebras, developed by Alejandro Bermejo, which includes the Algebra B and Treon Algebra. This algebras can define Lie and Malcev algebras when their product operations are derived from the Bermejo Algebras product. Central to this study are treons, complex entities that arise as isomorphisms of Algebra B when the field is real. We define the vector spaces associated with Bermejo Algebras to establish equivalence classes and a quotient topology, resulting in Hausdorff spaces that do not depend on traditional norms, inner products, or metrics. Our findings lay the groundwork for constructing new types of differential manifolds, opening new avenues for advanced mathematical research.

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