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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.

Kodiologist
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1 Answers1

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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

drhab
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