Homotopy of Linearly Ordered Split–Join Chains in Covering Spaces of Foliated n-Manifold Charts

Topological spaces can be induced by various algebraic ordering relations such as, linear, partial and the inclusion-ordering of open sets forming chains chain complexes. In general, classifications covering are made using fundamental groups lifting. However, Riesz ordered n-spaces Urysohn interpretations real-valued continuous functions as provide new perspectives. This paper proposes formulation n-space charts a foliated n-manifold containing linearly chains, where do not form topologically separated components within section. The chained subspaces subjected to split–join operations under bijective function chain-subspaces simply directed twisted chains. resulting chain-paths oriented homotopy path-products involving function. It is shown that embedding any in leaf homogeneous unique. finite measures topological homotopies generate multiplicative cyclic group varieties different orders depending upon types measures. As distinction, proposed consider formation circular nerves Nachbin preordering, thereby avoiding symmetry/asymmetry conditions.