Uniform structure induces topology

Statement
Given a fact about::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 fact about::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.