Remark

If $f : X \to Y$ is surjective, then the entourages of $Y$ are the direct images under $g$ of the entourages of $X$.

A mapping $f$ of a uniform space $X$ into a uniform space $X'$ is uniformly continuous if and only if the inverse image under $f$ of the uniformity of $X'$ is coarser than the uniformity of $X$.

Let $A$ be a subset of a uniform space $X$. The uniformity induced on $A$ by the uniformity of $X$ is the inverse image of the latter under the canonical injection $A \to X$. By Proposition 4 of no. 3, this is equivalent to the following definition: