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.
It is known from measurements of beat frequencies of beam pairs extracted from single mode dual polarized ring
lasers that the optical cavity can support four different phase velocities depending on left or right polarization
and on clockwise or anticlockwise direction of travel. The topological theory of singular solutions to Maxwell's
equations has demonstrated that these four modes are due to different Spinor pairs. Moreover, the theory
demonstrates that the Lorentz Vacuum and the Chiral Vacuum can be formally indistinguishable, except for
the impedance of free space (which can related to the determinant of a dynamic constitutive tensor and therefore
to the chiral polarization and expansion coefficients involving Optical Activity and Faraday rotation). It is
suggested that modifications of the dual polarized Sagnac ring laser with its resonance Q of 10<sup>+18</sup> might permit
the experimental detection of any chiral properties of space-time induced by intransitive motions (those with
fixed points of rotation and expansion) that are not induced by transitive motions (translations).
At its foundations, Maxwell's theory of Electrodynamics, like thermodynamics, is a topological theory independent from geometric constraints of metric, scales, or gauge symmetries. One of the most interesting features of Electromagnetism is its relationship to the transport of momentum and energy by means of photons. This article utilizes a topological perspective to discuss the classical features and quantum concepts associated with the photon, including Topological Spin, Topological Torsion, Helicity and Chirality.