A Solution of Problem 540

Problem 540 of J. D. Lawson and M. Mislove in Open Problems in
Topology asks whether the process of taking duals terminate after finitely many steps with
topologies that are duals of each other. The problem was for $T_1$ spaces already
solved by G. E. Strecker in 1966. For certain topologies on hyperspaces
(which are not necessarily $T_1$), the main question was in the positive answered by Bruce S. Burdick
and his solution was presented on The First Turkish International Conference on Topology in Istanbul in 2000.
In this paper we bring a complete and positive solution of the problem for all topological
spaces. We show that for any topological space $(X,\tau)$ it follows
$\tau^{dd}=\tau^{dddd}$. Further, we classify topological spaces with respect to
the number of generated topologies by the process of taking duals.