Exercise 1, § 3

Let $\mathfrak{U}$ denote the additive uniformity on the real line $\mathbf{R}$, and let $\mathfrak{U}'$ denote the inverse image of $\mathfrak{U}$ under the mapping $x \to x^3$ of $\mathbf{R}$ onto itself. Show that $\mathfrak{U}'$ is strictly finer than $\mathfrak{U}$, but that the Cauchy filters are the same for both uniformities.*