Iterované dualizace topologických struktur

On iterated dualizations of topological structures

conference paper

en

A topology $\tau^d$ is said to be dual with respect to the topology $\tau$ on a set $X$ if $\tau^d$ has a closed base consisting of the compact saturated sets in the topological space $(X,\tau)$. In the well-known book{\it Open Problems in Topology}, edited by J. van Mill and G. M. Reed, there was stated the problem no. 540 of J. D. Lawson and M. Mislove: {\it Does the process of iterating duals of a topology terminate by two topologies, dual to each other (1990, \cite{LM})?} As a matter of fact, for $T_1$ spaces, the problem was solved by G. E. Strecker, J. de Groot and E. Wattel (1966, \cite{GSW}) a long time before it was formulated by Lawson and Mislove, since in $T_1$ spaces, the dual operator studied by Lawson and Mislove coincides with another dual, introduced by de Groot, Strecker and Wattel more than 30 years ago. In 2000 the problem was partially solved by B. Burdick, who proved that for some topologies on certain hyperspaces, during the iterated dualization process there can arise at most four distinct topologies: the original topology $\tau$, then $\tau^d$, $\tau^{dd}$ and $\tau^{ddd}$. Finally, this result was generalized for all topological spaces by M. M. Kov\' ar (2001,\cite{Ko}). In this talk we will speak about the following rings of questions: \roster \item We will present some recent and hot results related to iterated dualizations of topological spaces. \item We will ask what happens with the dualizations if we leave the realm of spatiality. \item We will mention some (unsolved) problems related to dual topologies. \endroster

Práce obsahuje některé nové výsledky o vlastnostech zobecněných de Grootových dualizací obecných topologických struktur. Jsou zmíněny a připomenuty také některé aktuální, nevyřešené problémy.

A topology $\tau^d$ is said to be dual with respect to the topology $\tau$ on a set $X$ if $\tau^d$ has a closed base consisting of the compact saturated sets in the topological space $(X,\tau)$. In the well-known book{\it Open Problems in Topology}, edited by J. van Mill and G. M. Reed, there was stated the problem no. 540 of J. D. Lawson and M. Mislove: {\it Does the process of iterating duals of a topology terminate by two topologies, dual to each other (1990, \cite{LM})?} As a matter of fact, for $T_1$ spaces, the problem was solved by G. E. Strecker, J. de Groot and E. Wattel (1966, \cite{GSW}) a long time before it was formulated by Lawson and Mislove, since in $T_1$ spaces, the dual operator studied by Lawson and Mislove coincides with another dual, introduced by de Groot, Strecker and Wattel more than 30 years ago. In 2000 the problem was partially solved by B. Burdick, who proved that for some topologies on certain hyperspaces, during the iterated dualization process there can arise at most four distinct topologies: the original topology $\tau$, then $\tau^d$, $\tau^{dd}$ and $\tau^{ddd}$. Finally, this result was generalized for all topological spaces by M. M. Kov\' ar (2001,\cite{Ko}). In this talk we will speak about the following rings of questions: \roster \item We will present some recent and hot results related to iterated dualizations of topological spaces. \item We will ask what happens with the dualizations if we leave the realm of spatiality. \item We will mention some (unsolved) problems related to dual topologies. \endroster

compact saturated set, dual topology, topological system, frame, locale, preframe

2002

24.06.2002

Shimane University in Matsue Osaka university

Matsue, Japonsko

Abstract of the International Conference on Topology and Its Applications - Topology in Matsue

1

2