  Let T be a topology on the finite set Xn. We consider the open polynomial
associated with the topology T. Its coefficients are the cardinality of open
sets of size j=0,...,n. J. Brown [4] asked when this polynomial has only real
zeros. We prove that this polynomial has real zeros, only in the trivial case
where T is the discrete topology. Then, we weaken Brown's question: for which
topology this polynomial is log-concave, or at least unimodal? A partial answer
is given. Precisely, we prove that if the topology has a large number of open
sets, then its open polynomial is unimodal.
