Homology under monotone maps between finite topological spaces

It is shown that a surjective monotone map $X\to Y$ between finite $T_0$-spaces induces a surjective map on homology. As such a map turns out to be a sequence of edge contractions in the Hasse diagram of $X$, followed by a homeomorphism, this leads to an explicit relation between the Betti numbers of $X$ to those of $Y$ and the cokernels of the edge contraction maps on the order complexes.