Hyperconnected Relator Spaces. CW Complexes and Continuous Function Paths that are Hyperconnected

Abstract: This article introduces proximal cell complexes in a hyperconnected space. Hyperconnectedness encodes how collections of path-connected sub-complexes in a Alexandroff-Hopf-Whitehead CW space are near to or far from each other. Several main results are given, namely, a hyper-connectedness form of CW (Closure Finite Weak topology) complex, the existence of continuous functions that are paths in hyperconnected relator spaces and hyperconnected chains with overlapping interiors that are path graphs in a relator space. An application of these results is given in terms of the definition of cycles using the centroids of triangles.