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.