We consider those Hausdorff continua S such that each separable subspace of S is hereditarily separable.Due to results of Ostaszewski and Rudin, respectively, all monotonically normal spaces trucf and therefore all continuous Hausdorff images of ordered compacta also have this property.Our study focuses on the structure cath kidston forget me not of such spaces that also possess one of various rim properties, with emphasis given to rim-separability.In so doing we obtain analogues of results of M.Tuncali and I.
Loncar, respectively.