Does there always exist a refinement of a partition that is regular?

Question

The title is pretty explanatory, but more formally, I'm wondering whether given a partition $P$ of a closed interval $[a,b]$, there exists a refinement $P^*$ such that $P^*$ is regular.

Thanks!

Answer 1

If by regular partition you mean what I would call a uniform partition, one in which the subintervals all have the same length, then the answer is no. Take $a=0$ and $b=3$, and let $P$ be the partition $x_0=0,x_1=1,x_2=\sqrt2,x_3=3$. Suppose that there were a uniform partition with interval size $\Delta$ that refined $P$; then there would be positive integers $n$ and $m$ such that $n\Delta=1$ and $m\Delta=\sqrt2$. But then we’d have $\sqrt2=\frac{m}n$, making $\sqrt2$ rational, which is of course false.

You’ll run into the same problem with any partition that has subintervals of both rational and irrational lengths.