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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!

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Yes. Sure. $\,$ –  Berci Feb 24 '13 at 2:41
    
@Berci: How is it done? –  Tim Feb 24 '13 at 2:41

1 Answer 1

up vote 3 down vote accepted

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.

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Thanks! (1) Wikipedia defines a mapping $f:X\to Y$ between uniform spaces $X$ and $Y$ to be uniformly continuous, if for every entourage $V$ in $Y$, there exists an entourage $U$ in $X$ s.t. $(f \times f)(U) \subseteq V$, i.e. for every $(x_1, x_2) \in U$ we have $(f(x_1), f(x_2)) \in V$. I was wondering if that is equivalent to the definition you gave? –  Tim Feb 24 '13 at 2:57
    
(2) Can $\{ (f\times f)^{-1} (U), \forall U \in M\}$ be a uniform structure on $X$ without any further operation? If I am correct, given a mapping $f$ from a set $X$ to a topological space $Y$ with topology $\tau$, the smallest topology on $X$ s.t. $f$ is continuous is $\{f^{-1}(O), \forall O \in \tau\}$? –  Tim Feb 24 '13 at 3:01
1  
As for question 1, it's the same. The wiki definition implies the condition I stated simply because it implies that $U \subseteq (f\times f)^{-1}(V)$, and entourages are upwards closed. As for 2, I don't know this off the top of my head. Checking should be very easy, just verify if the given collection is a uniform structure. –  Ittay Weiss Feb 24 '13 at 3:14
    
Thanks! For 2, I think it is not a uniform structure yet, because it may not be upwards closed, and the diagonal may not be an entourage. Am I right? –  Tim Feb 24 '13 at 5:14

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