Uniform structure induced by a mapping from its codomain to its domain?

Question

Suppose $X$ is a set and $Y$ is a uniform space with uniform structure $M$.

1. Given a mapping $f: X \to Y$, I was wondering if $f$ can induce a uniform structure on $X$ from the uniform structure $M$ on $Y$, s.t. $f$ is uniformly continuous?

   Is the induced uniform structure on $X$ the smallest one s.t. $f$ is uniformly continuous?

2. Given a family of mappings $\{f_\alpha: X\to Y, \alpha \in I\}$, how do they induce a uniform structure on $X$, s.t. the mappings are all uniformly continuous?

   Is the induced uniform structure on $X$ the smallest one s.t. the mappings are uniformly continuous?

Thanks and regards!

Answer 1

Recall that a function $f:A \to B$ between uniform spaces is uniform precisely when $(f\times f)^{-1}(U)$ is an entourage in $A$ for all entourages $U$ in $B$.

So, if you want your $f:X\to Y$ to induce a uniform structure on $X$ from the one on $Y$, such that it is the smallest uniform structure such that $f$ is uniform, then you define the uniform structure on $B$ generated by all sets $(f\times f)^{-1}(U)$, where $U$ ranges over the entourages in $Y$. It is immediate that this collection of subsets of $X\times X$ forms a base for a uniform structure on $X$, clearly having the desired property.

The answer for a family of functions is essentially the same, just involved some more indices.