Sequence of dualizations of topological spaces is finite

Details Sequence of dualizations of topological spaces is finite Sequence of dualizations of topological spaces is finite

Publishing date

2002-04-10

URL

arxiv.org/abs/math/0204132v1

Journals

Proceedings of the Ninth Prague Topological Symposium, (Prague, 2001), pp. 171--179, Topology Atlas, Toronto, 2002}

Science

Mathematics

Field

General Topology

Additional info

9 pages. Reprinted from Theoretical Computer Science B, in press, Martin Maria Kovar, The solution to Problem 540

Type

Scientific paper

Abstract

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 $T_1$ spaces was 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.

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