At most 4 topologies can arise from iterating the de Groot dual

Problem 540 of J.D. Lawson and M. Mislove in Open Problems in Topology asks whether the process of taking duals terminates after finitely many steps with topologies that are duals of each other. The problem for T1 spaces was solved by G.E. Strecker in 1966. For certain topologies on hyperspaces (which are not necessarily T1), the main question was answered in the positive 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 show that for any topological space (X,τ) it follows τdd=τdddd. Further, we classify topological spaces with respect to the number of generated topologies by the process of taking duals.