On spaces in which every bounded subset is Hausdorff (Q2640180)

Denote by Haus(comp) (or Haus(e-comp)) the class of topological spaces X in which every compact (or e-compact, resp.) subspace is Hausdorff, where e-compactness for Y means the existence of a dense subset B in Y such that every open cover of Y contains a finite cover of B. The paper contains examples that the inclusion Haus(e-comp)\(\subset Haus(comp)\) is proper and does not preserve epimorphisms, and that Haus(e-comp) is not co-well-powered [the fact that Haus(comp) is not co-well-powered was proved by the first author and the reviewer in Ann. Mat. Pura Appl., IV. Ser. 145, 337-346 (1986; Zbl 0617.54006)]. To prove the last assertion, the Haus(e-comp)-closure is described as the idempotent hull of a modification of the compactly generated closure.