Different Uniformities generating Same Topology

Question

Can anyone please give me example of different uniformities that induces the same topology? I came to know that such uniformities exist via online, but I failed to construct one.

Answer 1

For instance, the discrete topology on the set $\Bbb Z$ or $\Bbb R$ can be generated by a discrete uniformity with a base $\{\Delta\}$ or by a uniformity with a base $\{U_n:n\in\Bbb N\}$, where $U_n=\{(x,y): x=y$ or $x,y\ge n\}$.

Answer 2

In fact almost any uniform space is an example: a Tychonoff (so uniformisable) space $X$ is called "almost compact" if there is a unique uniformity that induces its topology. It turns out the there are the spaces $X$ where $\beta(X)\setminus X$ has at most one point. It includes the compact (Hausdorff) spaces, and also spaces like $\omega_1$. So e.g. $\mathbb{R}$ and $\mathbb{N}$ have plenty such different uniformities.