Dual spaces of some congruence lattices

For any set X and any n⩾3 we define a topological space Ln(X) and characterize its closed subspaces if |X|⩽ℵ1. As an application we obtain a characterization of congruence lattices of algebras in some varieties of lattices. The spaces Ln(X) are close to Boolean spaces, but they are not Hausdorff.