Coarse metric approximation

We generalize the concept of coarsely n-to-1 maps by introducing coarsely finite-to-1 maps and show that the new concept is a defining property of large scale finitistic spaces. Furthermore, we introduce an approximation of large scale (also called coarse) spaces by metric spaces, called the coarse metric approximation, and develop the corresponding theory. The mentioned approximation is a tool which allows us to generalize statements about coarse properties of metric spaces to equivalent statements about general large scale spaces. As a result we generalize the coarse versions of the Dimension Raising Theorem, Finite-To-One Mapping Theorem and the obtained characterization of large scale finitistic spaces to the case of general coarse structures.