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Let $d_1$ and $d_2$ be two metrics on the same set $X$. Then $d_1$ and $d_2$ are uniformly equivalent if the identity maps $i:(M,d_1)\rightarrow(M,d_2)$ and $i^{-1}:(M,d_2)\rightarrow(M,d_1)$ are uniformly continuous. My question is, are two metric spaces uniformly equivalent if and only if they have the same zero distance sets? That is, $d_1(A,B)=0$ if and only if $d_2(A,B)=0$ for all subsets $A$ and $B$ of $M$.

Or to put it in fancier language, do two metrics induce the same uniformity if and only if they induce the same proximity? If not, which direction is false?

For those who don’t know, $d_1(A,B)=\inf\{d_1(a,b):a\in A, b\in B\}$, and similarly for $d_2$.

José Carlos Santos
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Keshav Srinivasan
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1 Answers1

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Yes. I found the result stated as an exercise in this excerpt from Ryszard Engelking's book General Topology:

8.5.19. A. Let $(X,\rho)$ and $(Y,\sigma)$ be metric spaces and let $\delta$ and $\delta'$ denote the proximities induced by $\rho$ and $\sigma$ on $X$ and $Y$ respectively. Show that a mapping $f$ of $X$ to $Y$ is proximally continuous if and only if $f$ is uniformly continuous with respect to $\rho$ and $\sigma$.
B. Note that the metric[s] $\rho_1$ and $\rho_2$ on a set $X$ are uniformly equivalent if and only if they induce the same proximity.

Here proximally continuous means that the images of two sets which are close with respect to $\rho$ are close with respect to $\sigma$.

So two metrics induce the same uniformity if and only if they induce the same proximity. Which I think means that metrizable uniformities and metrizable proximities can be viewed as fundamentally the same structure.

Keshav Srinivasan
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