  There exists a physically well motivated method for approximating manifolds
by certain topological spaces with a finite or a countable set of points. These
spaces, which are partially ordered sets (posets) have the power to effectively
reproduce important topological features of continuum physics like winding
numbers and fractional statistics, and that too often with just a few points.
In this work, we develop the essential tools for doing quantum physics on
posets. The poset approach to covering space quantization, soliton physics,
gauge theories and the Dirac equation are discussed with emphasis on physically
important topological aspects. These ideas are illustrated by simple examples
like the covering space quantization of a particle on a circle, and the
sine-Gordon solitons.
