Uniform structure induces topology

Statement

Given a Uniform space (?) $(X,\mathcal{U})$ (i.e., a set $X$ with a uniform structure $\mathcal{U}$) we define a topology on $X$ as follows (thus turning $X$ into a Topological space (?)): A subset $V\subseteq X$ is said to be open if, for every $x\in V$, there exists $U\in \mathcal{U}$ such that whenever $(x,y)\in U$, we have $y\in V$.

Often, when we talk of a uniform structure on a topological space, we mean a uniform structure whose induced topology is the given topology.