Surjective Identifications of Convex Noetherian Separations in Topological (C, R) Space

Abstract

The interplay of symmetry of algebraic structures in a space and the corresponding topological properties of the space provides interesting insights. This paper proposes the formation of a predicate evaluated P-separation of the subspace of a topological (C, R) space, where the P-separations form countable and finite number of connected components. The Noetherian P-separated subspaces within the respective components admit triangulated planar convexes. The vertices of triangulated planar convexes in the topological (C, R) space are not in the interior of the Noetherian P-separated open subspaces. However, the P-separation points are interior to the respective locally dense planar triangulated convexes. The Noetherian P-separated subspaces are surjectively identified in another topological (C, R) space maintaining the corresponding local homeomorphism. The surjective identification of two triangulated planar convexes generates a quasiloop–quasigroupoid hybrid algebraic variety. However, the prime order of the two surjectively identified triangulated convexes allows the formation of a cyclic group structure in a countable discrete set under bijection. The surjectively identified topological subspace containing the quasiloop–quasigroupoid hybrid admits linear translation operation, where the right-identity element of the quasiloop–quasigroupoid hybrid structure preserves the symmetry of distribution of other elements. Interestingly, the vertices of a triangulated planar convex form the oriented multiplicative group structures. The surjectively identified planar triangulated convexes in a locally homeomorphic subspace maintain path-connection, where the right-identity element of the quasiloop–quasigroupoid hybrid behaves as a point of separation. Surjectively identified topological subspaces admitting multiple triangulated planar convexes preserve an alternative form of topological chained intersection property.