Iterované dualizace topologických prostorů a struktur

On iterated dualizations of topological spaces and structures

conference paper

en

Recall that 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 consisted of the compact saturated sets in $\tau$. In the well-known book{\it Open Problems in Topology}, edited by J. van Mill and G. M. Reed, there was stated (among many others, no less interesting problems) a 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})?} In this paper we will present some recent results related to iterated dualizations of topological spaces (one of them yields the above mentioned identity $\tau^{dd}=\tau^{dddd}$ as an immediate consequence), ask what happens with the dualizations if we leave the realm of spatiality and mention some unsolved problems related to dual topologies.

V práci jsou dokázány další výsledky o de Grootově duálu topologického prostoru, ale i dalších struktur. Zejména je dokázána nová a obecnější identita o duálních topologiích, z níž vyplývají mj. i některé již publikované výsledky.

Recall that 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 consisted of the compact saturated sets in $\tau$. In the well-known book{\it Open Problems in Topology}, edited by J. van Mill and G. M. Reed, there was stated (among many others, no less interesting problems) a 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})?} In this paper we will present some recent results related to iterated dualizations of topological spaces (one of them yields the above mentioned identity $\tau^{dd}=\tau^{dddd}$ as an immediate consequence), ask what happens with the dualizations if we leave the realm of spatiality and mention some unsolved problems related to dual topologies.

compact saturated set, dual topology, topological system, frame, locale, directly complete semilattice

2002

03.05.2002

City College, City University of New York

New York, Spojené státy americké

Abstracts of the Workshop on Topology in Computer Science

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