A Super-Multiplicative Inequality for the Number of Finite Unlabeled and T0 Topologies

Let n be a nonnegative integer, and let f(n) denote the number of unlabeled finite topologies on n points. We show that the inequality f(n+m)≥f(n)f(m) holds both for both labeled and unlabeled finite topologies. In addition, we establish a similar inequality for labeled and unlabeled T0 topologies.