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 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.

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. `http://www.elsevier.com/locate/tcs` `http://www.sciencedirect.com`

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