Complete lattices of subgroups in compact groups

Summary LetG be a compact group and a sublattice of the lattice of all closed subgroups ofG. In Proposition 1 it is shown that is a complete lattice if it is a closed subset of the spaceG ^c of all closed non empty subsets ofG. In general the converse of this fact is not true (Example 3), but the following result can be obtained (Theorem 5): If is complete and if each element of is normalized by the connected component of the identity ofG, then is a closed, totally disconnected subset ofG ^c. We mention the following corollary: IfG is totally disconnected or abelian, then is complete if and only if it is a closed subset ofG ^c.While writing this paper the author was a fellow of the National Research Council (A 7171).