Exercise 4, § 2

a) Let $\mathfrak{U}$ and $\mathfrak{U}'$ be the filters of entourages of two uniformities on the same non-empty set $X$. Show that the intersection of the filters $\mathfrak{U}$ and $\mathfrak{U}'$ is the filter of entourages of a uniformity on $X$ if and only if, for each $V \in \mathfrak{U}$ and $V' \in \mathfrak{U}'$, there exist $W \in \mathfrak{U}$ and $W' \in \mathfrak{U}'$ such that $WW' \subset V \cup V'$.

b) Give an example of two filters of entourages $\mathfrak{U}_{\overline{\omega}}, \mathfrak{U}_{\overline{\omega}'}$, each defined by a finite partition of $X$ (no. 5, Example), which do not satisfy the condition of a.