uniform structure of a metric space

Let $(X,d)$ be a metric space. There is a natural uniform structure on $X$ , which induces the same topology as the metric. We define a subset $V$ of the Cartesian product $X \times X$ to be an entourage if and only if it contains a subset of the form$$ V_{\varepsilon} = \{ (x,y) \in X \times X : d(x,y) < \varepsilon\}$$ for some $\varepsilon$ .