Publication detail

A Solution of Problem 540

KOVÁR, M.

Original Title

A Solution of Problem 540

English Title

A Solution of Problem 540

Type

journal article - other

Language

en

Original Abstract

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.

English abstract

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.

Keywords

saturated set, dual topology, compactness operator

RIV year

2002

Released

06.06.2001

Pages from

1

Pages to

7

Pages count

7

Documents

BibTex


@article{BUT40137,
  author="Martin {Kovár}",
  title="A Solution of Problem 540",
  annote="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.",
  chapter="40137",
  journal="Topology Atlas, Questions & Answers, Topology Atlas Document # idec-33, http://at.yorku.ca/i/d/e/c/33.htm",
  number="470",
  volume="2001",
  year="2001",
  month="june",
  pages="1",
  type="journal article - other"
}