The Non-Equivalence of $\tau$-Ultracompactness and $\tau$-Boundedness

Buras Boljiev

The main result presented here is a solution to the following problem of V. Saks: Does there exist $\mathfrak M>\aleph_0$ and a Hausdorff $\mathfrak M$-ultracompact space which is not $\mathfrak M$-bounded? The main result is given in a stronger form than the problem suggests itself: For each infinite cardinal $\tau$ there is a Hausdorff $\tau$-ultracompact not $\tau$-bounded space of density $\tau$.