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

Type

journal article - other

Language

English

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.

Keywords

saturated set, dual topology, compactness operator

Authors

KOVÁR, M.

RIV year

2003

Released

1. 5. 2003

ISBN

0166-8641

Periodical

Topology and its Applications

Year of study

2003

Number

130

State

Kingdom of the Netherlands

Pages from

175

Pages to

182

Pages count

8

BibTex

@article{BUT41534,
  author="Martin {Kovár}",
  title="At most 4 topologies can arise from iterating the de Groot dual",
  journal="Topology and its Applications",
  year="2003",
  volume="2003",
  number="130",
  pages="175--182",
  issn="0166-8641"
}