Collectionwise Hausdorff: incompactness at singulars

Under set theoretic hypotheses, we construct a λ-collectionwise Hausdorff not λ⁺- collectionwise Hausdorff space of character c for certain singular cardinals λ. For example if V = L, and cf(λ) is not weakly compact, or if there are no inner models with large cardinals, λ is singular strong limit, and cf(λ) is the successor of a singular strong limit. Moreover, after forcing collapsing c to ω these spaces retain their properties; thus we obtain first countable examples.