Abstract
The algebraic as well as geometric topological constructions of manifold embeddings and homotopy offer interesting insights about spaces and symmetry. This paper proposes the construction of 2-quasinormed variants of locally dense p-normed 2-spheres within a non-uniformly scalable quasinormed topological (C, R) space. The fibered space is dense and the 2-spheres are equivalent to the category of 3-dimensional manifolds or three-manifolds with simply connected boundary surfaces. However, the disjoint and proper embeddings of covering three-manifolds within the convex subspaces generates separations of p-normed 2-spheres. The 2-quasinormed variants of p-normed 2-spheres are compact and path-connected varieties within the dense space. The path-connection is further extended by introducing the concept of bi-connectedness, preserving Urysohn separation of closed subspaces. The local fundamental groups are constructed from the discrete variety of path-homotopies, which are interior to the respective 2-spheres. The simple connected boundaries of p-normed 2-spheres generate finite and countable sets of homotopy contacts of the fundamental groups. Interestingly, a compact fibre can prepare a homotopy loop in the fundamental group within the fibered topological (C, R) space. It is shown that the holomorphic condition is a requirement in the topological (C, R) space to preserve a convex path-component. However, the topological projections of p-normed 2-spheres on the disjoint holomorphic complex subspaces retain the path-connection property irrespective of the projective points on real subspace. The local fundamental groups of discrete-loop variety support the formation of a homotopically Hausdorff (C, R) space.
Highlights
A path-connected topological space is considered to be locally path-connected within a path-component maintaining the equivalence relation
A q-quasinormed topological space can admit a corresponding topology generated by the respective p-norm function
The proposed constructions of 2-quasinormed variety of locally dense p-normed 2-spheres within a non-uniformly scalable quasinormed topological (C, R) space enable the formulation of path-connected fundamental groups interior to it
Summary
A path-connected topological space is considered to be locally path-connected within a path-component maintaining the equivalence relation. A first countable path-connected topological space admits countable fundamental groups if the space is a homotopically Hausdorff variety [1]. In a one-dimensional topological space X the fundamental group π(X) becomes a free group if the space is a connected type [1] In this case the topological space successfully admits a suitable metric structure. This paper proposes the topological construction and analysis of 2-quasinormed variants of p-normed 2-spheres, path-connected fundamental groups and associated homotopy contacts in a fibered as well as quasinormed topological (C, R) space [3]. The space is non-uniformly scalable and the fundamental groups are interior to dense subspaces of 2-quasinormed variant of p-normed 2-spheres generating a set of homotopy contacts. The surfaces of three-manifolds and 2-spheres are often alternatively named as respective boundaries for the simplicity of presentation
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