Spectral subspaces of spectra of Abelian lattice-ordered groups in size aleph one

International audience ; It is well known that the lattice Idc G of all principal ℓ-ideals of any Abelian ℓ-group G is a completely normal distributive 0-lattice, and that not every completely normal distributive 0-lattice is a homomorphic image of some Idc G, via a counterexample of cardinality ℵ 2. We prove that every completely normal distributive 0-lattice with at most ℵ 1 elements is a homomorphic image of some Idc G. By Stone duality, this means that every completely normal generalized spectral space, with at most ℵ 1 compact open sets, is homeomorphic to a spectral subspace of the ℓ-spectrum of some Abelian ℓ-group.