About: A Solution of Problem 540

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Description	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 was for $T_1$ spaces 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. 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 was for $T_1$ spaces 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. (en)
Title	A Solution of Problem 540 A Solution of Problem 540 (en)
skos:prefLabel	A Solution of Problem 540 A Solution of Problem 540 (en)
skos:notation	RIV/00216305:26220/01:PU22440!RIV11-MSM-26220___
http://linked.open...avai/riv/aktivita	N P V Z
http://linked.open...avai/riv/aktivity	N, P(GA201/00/1466), V, Z(MSM 262200012)
http://linked.open...iv/cisloPeriodika	470
http://linked.open...vai/riv/dodaniDat	2011
http://linked.open...aciTvurceVysledku	Kovár, Martin
http://linked.open.../riv/druhVysledku	J - Článek v odborném periodiku
http://linked.open...iv/duvernostUdaju	S - Úplné a pravdivé údaje nepodléhající ochraně podle zvláštních právních předpisů
http://linked.open...titaPredkladatele	Vysoké učení technické v Brně / Fakulta elektrotechniky a komunikačních technologií
http://linked.open...dnocenehoVysledku	672338
http://linked.open...ai/riv/idVysledku	RIV/00216305:26220/01:PU22440
http://linked.open...riv/jazykVysledku	eng - angličtina
http://linked.open.../riv/klicovaSlova	saturated set, dual topology, compactness operator (en)
http://linked.open.../riv/klicoveSlovo	compactness operator dual topology saturated set
http://linked.open...odStatuVydavatele	CZ - Česká republika
http://linked.open...ontrolniKodProRIV	[90D3DB5CDD96]
http://linked.open...i/riv/nazevZdroje	Topology Atlas, Questions & Answers, Topology Atlas Document # idec-33, http://at.yorku.ca/i/d/e/c/33.htm
http://linked.open...in/vavai/riv/obor	BA
http://linked.open...ichTvurcuVysledku	1 (xsd:int)
http://linked.open...cetTvurcuVysledku	1 (xsd:int)
http://linked.open...vavai/riv/projekt	Continuous and set-theoretical methods in topological and algebraic structures
http://linked.open...UplatneniVysledku	2001
http://linked.open...v/svazekPeriodika	2001
http://linked.open...iv/tvurceVysledku	Kovár, Martin
http://linked.open...n/vavai/riv/zamer	http://linked.opendata.cz/resource/domain/vavai/zamer/MSM%20262200012
number of pages	7 (xsd:int)
http://localhost/t...ganizacniJednotka	26220

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