Different uniform spaces having the same set of Cauchy filters

Question

I want to understand how Cauchy space is different than uniform space. For this I need an example:

An example of two different uniform spaces having for both of them the same set of Cauchy filters?

Answer 1

On $\mathbb{R}$, consider the two metrics

$$d_1(x,y) = \lvert x-y\rvert;\qquad d_2(x,y) = \lvert x^3 - y^3\rvert,$$

and the uniform structures $\mathscr{U}_1,\,\mathscr{U}_2$ induced by them.

$h \colon x \mapsto x^3$ is an isometry $(\mathbb{R},\mathscr{U}_2) \to (\mathbb{R},\mathscr{U}_1)$, and it is continuous in $\mathscr{U}_1$, so the two uniform structures induce the same topology. Since $h^{-1}$ is uniformly continuous as a map $(\mathbb{R},\mathscr{U}_1) \to (\mathbb{R},\mathscr{U}_1)$, the identity $\operatorname{id}_\mathbb{R} = h^{-1}\circ h \colon (\mathbb{R},\mathscr{U}_2) \to (\mathbb{R},\mathscr{U}_1)$ is uniformly continuous, hence $\mathscr{U}_2$ is finer than $\mathscr{U}_1$. But $\operatorname{id}_\mathbb{R} \colon (\mathbb{R},\mathscr{U}_1) \to (\mathbb{R},\mathscr{U}_2)$ is not uniformly continuous, so $\mathscr{U}_2$ is strictly finer than $\mathscr{U}_1$.

Yet, $\mathscr{U}_1$ and $\mathscr{U}_2$ have the same Cauchy filters:

Since $\mathscr{U}_2$ is finer, every $\mathscr{U}_2$ Cauchy filter is a fortiori a $\mathscr{U}_1$ Cauchy filter.

Conversely, a $\mathscr{U}_1$ Cauchy filter is convergent. Since every convergent filter in a uniform space is a Cauchy filter, and both structures induce the same topology, it follows that all $\mathscr{U}_1$ Cauchy filters are also $\mathscr{U}_2$ Cauchy filters.