A note on iterated duals of certain topological spaces

Problem 540 in Open Problems in Topology ( 1990) asks whether iterating the operation of taking the dual topology eventually leads to a mutually dual pair of topologies. We give an affirmative answer to Problem 540 for several classes of spaces. Some of the special cases covered are: any T, space (already solved in 1966 by Strecker), the lower Victoris topology on any hyperspace, the Scott topology for reverse inclusion on any hyperspace, and the upper Victoris topology on the hyperspace of a regular space. We find in all these cases that T-dd = T-(dddd), and therefore at most four distinct topologies, T-, T-d T-dd, T-ddd, can be created by iterating the dual operator starting with any one of these special cases. (C) 2002 Published by Elsevier Science B.V.