Remark

A family $(\mathfrak{U}_i)$ of uniformities on $X$ also has a greatest lower bound in the ordered set of all uniformities on $X$, namely the least upper bound of the set of all uniformities on $X$ which are coarser than each of the $\mathfrak{U}_i$ (such uniformities exist, since the set of all uniformities on $X$ has a least element). But (supposing $X$ not empty) the filter of entourages of this uniformity is not necessarily the intersection of the filters of entourages of the $\mathfrak{U}_i$, because this latter filter need not satisfy axiom $(U_{III})$ (Exercise 4).