On Polynomials Associated with Finite Topologies

Let τ be a topology on the finite set Xn. We consider the open-set polynomial associated with the topology τ. Its coefficients are the cardinalities of sets Uj=Uj(τ) of open sets of size j=0,…,n. We prove that this polynomial has only real zeros only in the trivial case where τ is the discrete topology. Hence, we answer a question raised by J. Brown. We give a partial answer to the question: for which topology is this polynomial log-concave, or at least unimodal? More specifically, we prove that if the topology has a large number of open sets, its open polynomial is unimodal. The idea of degree of log-concavity is introduced and it is shown to be limited for polynomials of non-trivial topologies. Furthermore, the maximum-sized topologies that omit open sets of given sizes are derived. Moreover, all topologies over n points with at least (3/8)2n open sets are proved to be unimodal, completing previous results.