Preservation and reflection of separation axioms by essentially Kolmogorov and Kolmogorov relations

This paper focuses on the Kolmogorov functor $K:\mathrm{Top}\to {\mathrm{Top}}_0$ and associated ideas. There are two main objectives: first, catalogue and prove those topological invariants which K both preserves and reflects, called “Hong” invariants; and second, give a step-by-step axiomatic foundation for K to analyze its remarkable success in having so many Hong invariants. Pursuing the second objective leads to “essentially Kolmogorov” (EK) relations, the family of which on a ground set forms a complete lattice ordered by inclusion; the diagonal relation Δ is the universal lower bound and the Kolmogorov relation K is the universal upper bound—typically there are many EK relations strictly between Δ and K. Though EK relations are significant weakenings of K, they enjoy the same success w.r.t. Hong invariants. Counterexamples clarify relationships between similar notions.