The non-equivalence of τ-ultracompactness and τ-boundedness

Abstract

The main result presented here is a solution to the following problem of V. Saks: Does there exist M > ℵ 0 and a Hausdorff M-ultracompact space which is not M-bounded? The main result is given in a stronger form than the problem suggests itself: For each infinite cardinal τ there is a Hausdorff τ-ultracompact not τ-bounded space of density τ.