On iterated de Groot dualizations of topological spaces

Problem 540 of J. D. Lawson and M. Mislove in Open Problems in Topology ask whether the process of taking (de Groot) duals terminate after finitely many steps with topologies that are duals of each other. The question was solved in the positive by the author in 2001. In this paper we prove a new identity for dual topologies: $\tau^d=
(\tau\vee\tau^{dd})^d$ holds for every topological space $(X,\tau)$. We also present a solution of another problem that was open till now -- we give an equivalent internal characterization of those
spaces for which $\tau=\tau^{dd}$ and we also characterize the spaces satisfying the identities
$\tau^d=\tau^{ddd}$, $\tau=\tau^{d}$ and $\tau^d=\tau^{dd}$.