Publication detail

At most 4 topologies can arise from iterating the de Groot dual

KOVÁR, M.

Original Title

At most 4 topologies can arise from iterating the de Groot dual

English Title

At most 4 topologies can arise from iterating the de Groot dual

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 for $T_1$ spaces was 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 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 for $T_1$ spaces was 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 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

2003

Released

01.05.2003

ISBN

0166-8641

Periodical

Topology and its Applications

Year of study

2003

Number

130

State

NL

Pages from

175

Pages to

182

Pages count

8

Documents

BibTex


@article{BUT41534,
  author="Martin {Kovár}",
  title="At most 4 topologies can arise from iterating the de Groot dual",
  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 for $T_1$ spaces was 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 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="41534",
  journal="Topology and its Applications",
  number="130",
  volume="2003",
  year="2003",
  month="may",
  pages="175--182",
  type="journal article - other"
}