Exercise 5, § 2

Let $(X_i)_{i \in I}$ be a family of uniform spaces and let $c$ be an infinite cardinal. For each family $(V_i)_{i \in H}$, where card $(H) < c$ and $V_i$ is an entourage of $X_i$ for each $i \in H$, let $U((V_i))$ be the set of pairs $(x, y)$ of points of $X = \prod_{i \in I} X_i$ such that $(\mathrm{pr}_i x, \mathrm{pr}_i y) \in V_i$ for each $i \in H$. Show that the sets $U((V_i))$ form a fundamental system of entourages of a uniformity on $X$, and that the topology induced by this uniformity is that defined in Exercise 9 of Chapter I