Abstract
Since half a century, formulas for the enumeration of many classes of finite topological spaces have been found, though counting all topologies on an n-set remains a challenging open problem. In our paper, we give order-theoretical descriptions of specific classes of Alexandroff spaces such us submaximal, door, Whyburn and weakly Whyburn spaces. Connections with posets, counting formulas and numerical results in the class of finite spaces are also given.