Connected Fundamental Groups and Homotopy Contacts in Fibered Topological (C, R) Space

The algebraic as well geometric topological constructions of manifold embeddings and homotopy offer interesting insights about spaces symmetry. This paper proposes the construction 2-quasinormed variants locally dense p-normed 2-spheres within a non-uniformly scalable quasinormed (C, R) space. fibered space is are equivalent to category 3-dimensional manifolds or three-manifolds with simply connected boundary surfaces. However, disjoint proper covering convex subspaces generates separations 2-spheres. compact path-connected varieties path-connection further extended by introducing concept bi-connectedness, preserving Urysohn separation closed subspaces. local fundamental groups constructed from discrete variety path-homotopies, which interior respective simple boundaries generate finite countable sets contacts groups. Interestingly, fibre can prepare loop in group It shown that holomorphic condition requirement preserve path-component. projections on complex retain property irrespective projective points real subspace. discrete-loop support formation homotopically Hausdorff