All physical measurements are based on finite intervals of space and time. It follows that the appropriate topologies of measurement must be finite. However, there are only two types of finite power set topologies: T0 topologies and Not-T0 topologies. All singlet subsets of T0 (Kolmogorov) topologies are topologically distinguishable. Therefor it is natural that such topologies should be called Particle-like topologies. On the otherhand, some, if not all, singlet subsets of Not-T0 topologies are indistinguishable. Hence such topologies will be called Statistical, Wave-like, or Photon topologies. This article starts with a short review of the topological properties of Kolomogorov T0 particle topologies using processes that generate homotopic evolution of those exterior differential 1-forms chosed to describe thermodynamic states. Not-T0 topologies can use homotopic evolution of N-form densities to generate systems of partial differential equations that describe both reversible and irreversible dynamics. Numerous examples will be presented to demonstrate continuous topological evolution of complex exterior differential form densities in terms of Cartan’s homotopic magic formula.