Abstract
It is shown that the lattice of all closed subgroups of a compact topological group and the lattice of all connected closed subgroups of a pro-Lie group are algebraic, even arithmetic, if they are equipped with the order opposite to the natural one. The compact elements form ideals in these lattices and are explicitly determined. In the course of the proof the question is treated whether forming lattices of closed subgroups of topological groups commutes with projective limits.
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Scheiderer, C. Algebraic subgroup lattices of topological groups. Algebra Universalis 22, 235–243 (1986). https://doi.org/10.1007/BF01224029
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DOI: https://doi.org/10.1007/BF01224029