Abstract
In this paper we consider the lattice ΛG of all closed connected subgroups of pro-Lie groups G, which seems to have in some sense a more geometric nature than the full lattice of all closed subgroups. We determine those pro-Lie groups whose lattice shares one of the elementary geometric lattice properties, such as the existence of complements and relative complements, semi-modularity and its dual, the chain condition, self-duality and related ones. Apart from these results dealing with subgroup lattices we also get two structure theorems, one saying that maximal closed analytic subgroups of Lie groups actually are maximal among all analytic subgroups, the other that each connected abelian pro-Lie group is a direct product of a compact group with copies of the reals.
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