Lattices of homomorphisms and pro-Lie groups

Arkady G. Leiderman, Mikhail G. Tkachenko

Research output: Contribution to journalArticlepeer-review

2 Scopus citations


Early this century K. H. Hofmann and S. A. Morris introduced the class of pro-Lie groups which consists of projective limits of finite-dimensional Lie groups and proved that it contains all compact groups, all locally compact abelian groups, and all connected locally compact groups and is closed under the formation of products and closed subgroups. They defined a topological group G to be almost connected if the quotient group of G by the connected component of its identity is compact. We show here that all almost connected pro-Lie groups as well as their continuous homomorphic images are R-factorizable and ω-cellular, i.e. every family of Gδ-sets contains a countable subfamily whose union is dense in the union of the whole family. We also prove a general result which implies as a special case that if a topological group G contains a compact invariant subgroup K such that the quotient group G/K is an almost connected pro-Lie group, then G is R-factorizable and ω-cellular. Applying the aforementioned result we show that the sequential closure and the closure of an arbitrary Gδ,Σ-set in an almost connected pro-Lie group H coincide.

Original languageEnglish
Pages (from-to)1-20
Number of pages20
JournalTopology and its Applications
StatePublished - 1 Dec 2016


  • Almost connected group
  • Lindelöf Σ-group
  • Pro-Lie group
  • R-factorizable group
  • Weak (strong) σ-lattice

ASJC Scopus subject areas

  • Geometry and Topology


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