Finite topologies for finite geometries

Abstract: Without leaving finite mathematics and using finite topological spaces only, we give a definition of homeomorphisms of finite abstract simplicial complexes or finite graphs. Besides exploring the definition in various contexts, we add some remarks like that the general Lefschetz formula works for any continuous map on any finite topological space. We also noted that any higher order Wu characteristic as well as their cohomology are topological invariants which are not homotopy invariants. Energy theorems allow to express these topological invariants in terms of interaction energies of local open sets.