Topology Atlas Document # ppae-17


Sequence of dualizations of topological spaces is finite

Martin Maria Kovár

Proceedings of the Ninth Prague Topological Symposium (2001) pp. 171-179

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, \tau) it follows \taudd=\taudddd. Further, we classify topological spaces with respect to the number of generated topologies by the process of taking duals.

Mathematics Subject Classification. 54B99 54D30 54E55.
Keywords. saturated set, dual topology, compactness operator.

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Comments. This article is reprinted from Theoretical Computer Science B, in press, Martin Maria Kovár, The solution to Problem 540, Copyright (2002), with permission from Elsevier Science.

Copyright © 2002 Charles University and Topology Atlas. Published April 2002.