Topological spaces with finite topologies and nonsymmetric pseudometrics

In the present paper we study the topological spaces with finite topologies using the notion of elementary open (closed) set. We describe the family of open (closed) sets in terms of subordination matrix and subordination diagram, and characterize finite topologies satisfying the separability axioms $T_0$, $T_1$, $T_2$, $T_3$, $T_4$. We prove that topological spaces with finite topologies are locally linear connected, and each elementary open (closed) subset is homotopically equivalent to a point. We demonstrate that a topology of topological group is finite if and only if the set of connected components is finite and the connected component of unit has trivial topology. Also we define the nonsymmetric pseudometric and three topologies generated by this pseudometric. We prove that each finite topology is determined by a nonsymmetric pseudometric with two values.