Problem 540 is (almost) solved

Recall that a set is said to be saturated if it is the intersection of open sets. By the dual
topology $\tau^d$ for a topological space $(X,\tau)$ we mean the topology on $X$ generated by
taking the compact saturated sets of $X$ as a subbase for closed sets.
The Problem 540 of J. D. Lawson and M. Mislove \cite{LM} in Open Problems in
Topology (J. van Mill, G. M. Reed, eds.,1990)
asks

\medskip
\roster
\item which topologies can arise as dual topologies

\smallskip
and
\smallskip

\item whether the process of taking duals terminate after finitely many steps with
the topologies that are duals of each other.
\endroster
\medskip

For $T_1$ spaces, the solution of (2) simply follows from the fact that in $T_1$ spaces
every set is saturated and hence the dual operator $d$ coincide
with the compactness operator $\rho $ of J. de Groot, G. E. Strecker and E. Wattel \cite{GSW}.
For more general spaces, the question (2) was partially answered by Bruce S. Burdick who
found certain classes of (in general, non-$T_1$) spaces for which the process of taking duals
of a topological space $(X,\tau)$
terminates by $\tau^{dd}=\tau^{dddd}$ -- the lower Vietoris topology on any hyperspace,
the Scott topology for reverse inclusion on any hyperspace, and the upper Vietoris topology
on the hyperspace of a regular space. B. Burdick presented his paper on The First Turkish
International Conference on Topology in Istanbul 2000 \cite{Bu}.


\medskip

In this talk a general (and positive) solution of (2) with a short classification of topological spaces
with respect to the number of distinct topologies generated by iterating duals will be
presented. Our main result is the following theorem:


\proclaim{Theorem} For every topological space $(X,\tau)$ it follows $\tau^{dd}=
\tau^{dddd}$.
\endproclaim

On the other hand, we remark that this result cannot be improved since there exist a $T_1$ space
$(X,\tau)$ generating four distinct topologies $\tau$, $\tau^d=\rho(\tau)$, $\tau^{dd}=\rho^2(\tau)$ and
$\tau^{ddd}=\rho^3(\tau)$ (see e.g. Example 8 of \cite{GHSW}
or Example 1 of \cite{Bu}).