Let $\mathcal X$ be a partition of $ℝ$ into non-degenerate intervals (i.e., intervals that aren't just points). Must $\mathcal X$ be countable?
It's clear to me that the answer is yes if we assume that there's a minimum length $m$ of all the elements of $\mathcal X$. Then we can just make another partition where the intervals are all the same length (which is obviously countable) and line it up with $\mathcal X$. But when the infimum of the interval lengths is 0, it's no longer obvious to me that the conjecture is true.
Yes.
There is a function $\nu:\mathbb Q\to\mathcal X$ with $x\in\nu(x)$.
This is a surjective function since since every element $X$ of $\mathcal X$ contains an element of $\mathbb Q$.
Combined with the countability of $\mathbb Q$ this leads to the conclusion that $\mathcal X$ is countable