sentences:  However, the spaces in which every convergent filter has a unique limit are precisely the Hausdorff spaces.In general we must select, for each element of the index set, an element of the nonempty set of limits of the projected ultrafilter base, and of course this uses AC.However, it also shows that the compactness of the product of compact Hausdorff spaces can be proved using (BPI), and in fact the converse also holds.