Consider a finite topological space X. By using the relationship between digraphs and finite topological spaces, a corresponding digraph, GX can be found. Then, completing the adjacency matrix for GX can be used to determine the complexity class (defined via Turing machines) of problems involving GX, and therefore of X as well. This talk will examine the complexity classes of problems in deciding whether spaces are self-complementary finite topological spaces, deformation retracts, and if a finite topological space is a T0-space.
