product of uniform spaces

Definition.

Let ${\{X_{\alpha }\}}_{\alpha \in I}$ be a nonempty family of uniform spaces. The is the weakest uniformity (http://planetmath.org/UniformitiesOnASetFormACompleteLattice) on the Cartesian product $X={\prod }_{\alpha \in I}{X}_{\alpha }$ making all the projection maps ${\pi }_{\alpha }:X\to X_{\alpha }$ uniformly continuous.