Finite Topologies

Some remarks about finite topological spaces. During Spring break, I have looked for (and found) the right fusion inequality for Wu characteristic, a higher characteristic. Last year, I had shown for simplicial cohomology that there is the fusion inequality b(U)+b(K) larger or equal than b(G) if U, K are an open-closed complementary pair of G. In higher characteristic, where simplices can interact,this is more complicated. The inequality is b(U)+b(K) + b(U,K) + b(K,U)+b(U,U) larger or equal than b(G). I had missed the b(U,U) part. The reason is that in the Alexandrov topology on a simpicial complex one can have discoint open sets U,V (actually even smallest open sets) and points x in U, y in V such that x and y intersect. This reflects the fact that all finite topologies which are of some interest are non-Hausdorff.
At the end, there are some pictures near Medford center above the mystic river.
March 17: here is the blog entry with the code