Publication detail

On iterated de Groot dualizations of topological spaces

KOVÁR, M.

Original Title

On iterated de Groot dualizations of topological spaces

English Title

On iterated de Groot dualizations of topological spaces

Type

journal article - other

Language

en

Original Abstract

Problem 540 of J. D. Lawson and M. Mislove in Open Problems in Topology ask whether the process of taking (de Groot) duals terminate after finitely many steps with topologies that are duals of each other. The question was solved in the positive by the author in 2001. In this paper we prove a new identity for dual topologies: $\tau^d= (\tau\vee\tau^{dd})^d$ holds for every topological space $(X,\tau)$. We also present a solution of another problem that was open till now -- we give an equivalent internal characterization of those spaces for which $\tau=\tau^{dd}$ and we also characterize the spaces satisfying the identities $\tau^d=\tau^{ddd}$, $\tau=\tau^{d}$ and $\tau^d=\tau^{dd}$.

English abstract

Problem 540 of J. D. Lawson and M. Mislove in Open Problems in Topology ask whether the process of taking (de Groot) duals terminate after finitely many steps with topologies that are duals of each other. The question was solved in the positive by the author in 2001. In this paper we prove a new identity for dual topologies: $\tau^d= (\tau\vee\tau^{dd})^d$ holds for every topological space $(X,\tau)$. We also present a solution of another problem that was open till now -- we give an equivalent internal characterization of those spaces for which $\tau=\tau^{dd}$ and we also characterize the spaces satisfying the identities $\tau^d=\tau^{ddd}$, $\tau=\tau^{d}$ and $\tau^d=\tau^{dd}$.

Keywords

saturated set, dual topology, compactness operator

RIV year

2005

Released

01.01.2005

Pages from

83

Pages to

89

Pages count

7

BibTex


@article{BUT46466,
  author="Martin {Kovár}",
  title="On iterated de Groot dualizations of topological spaces",
  annote="Problem 540  of  J. D. Lawson and M. Mislove  in Open Problems  in Topology ask whether the process of taking  (de Groot) duals  terminate after finitely many steps with topologies that are duals of each other. The question was solved in the positive by the author in 2001. In this paper we prove a new identity for dual topologies: $\tau^d=
(\tau\vee\tau^{dd})^d$ holds for every topological space $(X,\tau)$. We also present a solution of another problem that was open till now  -- we give an equivalent internal characterization of those
spaces for which $\tau=\tau^{dd}$ and we also characterize the spaces satisfying the identities
$\tau^d=\tau^{ddd}$, $\tau=\tau^{d}$ and $\tau^d=\tau^{dd}$.",
  chapter="46466",
  journal="Topology and its Applications, IF=0.364 (2004)",
  number="146-7",
  volume="1",
  year="2005",
  month="january",
  pages="83",
  type="journal article - other"
}