Topologies on the subgroup lattice of a compact group

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1986
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Topology and its Applications. 1986, 23(2), pp. 183-191. ISSN 0166-8641. Available under: doi: 10.1016/0166-8641(86)90040-4
Zusammenfassung

Let G be a compact topological group. The lattice ΣG of its closed subgroups is algebraic in the reversed order, hence is made a compact topological semilattice by its dual Lawson topology. A second natural order-compatible compact topology on ΣG arises from the usual topology on the set of closed subsets of G. These topologies are shown to coincide precisely if the identity component is central in G, but to be essentially different otherwise, since they also fail to satisfy natural weakenings of the equality condition. In the second part of the paper the groups G are determined in which one of the lattice operations of ΣG becomes continuous with respect to either one of these topologies; several different characterizations of these cases are also provided.

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Fachgebiet (DDC)
510 Mathematik
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compact groups, subgroup lattices, algebraic lattices, topological semilattices
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ISO 690SCHEIDERER, Claus, 1986. Topologies on the subgroup lattice of a compact group. In: Topology and its Applications. 1986, 23(2), pp. 183-191. ISSN 0166-8641. Available under: doi: 10.1016/0166-8641(86)90040-4
BibTex
@article{Scheiderer1986Topol-23338,
  year={1986},
  doi={10.1016/0166-8641(86)90040-4},
  title={Topologies on the subgroup lattice of a compact group},
  number={2},
  volume={23},
  issn={0166-8641},
  journal={Topology and its Applications},
  pages={183--191},
  author={Scheiderer, Claus}
}
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