Differential geometry is a powerful tool in various branches of science, especially in theoretical physics. Ordinary differential geometry requires differentiable manifolds. This research paper shows how concepts of differential geometry can also be applied to pure topological spaces. Such a theory is based on concepts like cohomology theory. It allows to define a curvature operator also on pure topological spaces without connection. The main advantage of this theory is that the only required information about the topological spaces is the structure of these spaces. A formulation of quantum gravity is also possible with this theory.

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