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In the present paper, we propose a new axiomatic approach to nonstandard analysis and its application to the general theory of spatial structures in terms of category theory. Our framework is based on the idea of internal set theory, while we make use of an endofunctor $\mathcal{U}$ on a topos of sets S together with a natural transformation $\upsilon$, instead of the terms as “standard”, “internal”, or “external”. Moreover, we propose a general notion of a space called $\mathcal{U}$-space, and the category $\mathcal{U}Space$ whose objects are $\mathcal{U}$-spaces and morphisms are functions called $\mathcal{U}$-spatial morphisms. The category $\mathcal{U}Space$, which is shown to be Cartesian closed, gives a unified viewpoint toward topological and coarse geometric structure. It will also be useful to further study symmetries/asymmetries of the systems with infinite degrees of freedom, such as quantum fields. Full article

In the current study, distance-based topological invariants, namely the Wiener number and the topological roundness index, were computed for graphenic tori and Klein bottles (named toroidal and Klein bottle fullerenes or polyhexes in the pre-graphene literature) described as closed graphs with N vertices and 3N/2 edges, with N depending on the variable length of the cylindrical edge L_C of these nano-structures, which have a constant length L_M of the Möbius zigzag edge. The presented results show that Klein bottle cubic graphs are topologically indistinguishable from toroidal lattices with the same size (N, L_C, L_M) over a certain threshold size L_C. Both nano-structures share the same values of the topological indices that measure graph compactness and roundness, two key topological properties that largely influence lattice stability. Moreover, this newly conjectured topological similarity between the two kinds of graphs transfers the translation invariance typical of the graphenic tori to the Klein bottle polyhexes with size L_C ≥ L_C, making these graphs vertex transitive. This means that a traveler jumping on the nodes of these Klein bottle fullerenes is no longer able to distinguish among them by only measuring the chemical distances. This size-induced symmetry transition for Klein bottle cubic graphs represents a relevant topological effect influencing the electronic properties and the theoretical chemical stability of these two families of graphenic nano-systems. The present finding, nonetheless, provides an original argument, with potential future applications, that physical unification theory is possible, starting surprisingly from the nano-chemical topological graphenic space; thus, speculative hypotheses may be drawn, particularly relating to the computational topological unification (that is, complexification) of the quantum many-worlds picture (according to Everett’s theory) with the space-curvature sphericity/roundness of general relativity, as is also currently advocated by Wolfram’s language unification of matter-physical phenomenology. Full article