Sequence of Dualizations of Topological Spaces Is Finite

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 for T1 spaces was already solved by G. E. Strecker in 1966. For certain topologies on hyperspaces (which are not necessarily T1), 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, τ) it follows τ = τ. Further, we classify topological spaces with respect to the number of generated topologies by the process of taking duals. 0. Definitions and Notation Through the paper we mostly use the standard topological notions. The main source of definitions of some newer concepts is the paper [5] or the book [6] in which the reader can find many interesting connections to modern, computer science oriented topology. In this paper, all topological spaces are assumed without any separation axiom in general. Let (X, τ) be a topological space. Recall that the preorder of specialization is a reflexive and transitive binary relation on X defined by x 6 y if and only if x ∈ cl {y}. This relation is antisymmetric and hence a partial ordering if and only if X is a T0 space. For any set A ⊆ X we denote ↑ A = {x : x > y for some y ∈ A} and ↓ A = {x : x 6 y for some y ∈ A} . 2000 Mathematics Subject Classification. 54B99, 54D30, 54E55.