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The problem reads as follows:

Prove that $[0,1)$ cannot be expressed as a countable union of disjoint closed intervals.

While I was able to solve the problem, I am interested in how to solve it differently. The problem appears in Terence Tao's Measure Theory. He suggests the following: "...assume for sake of contradiction that $[0, 1)$ is the union of infinitely many closed intervals, and conclude that $[0, 1)$ is homeomorphic to the middle thirds Cantor set, which is absurd". The only connection to Cantor set that I see is in the following sort of algorithm: enumerate the intervals. Choose $I_{1}$ and define $I_{1}' = [0,1)\backslash I_{1}$. Then choose intervals $I_{2}$ and $I_{3}$ to the right and to the left of the interval $I_{1}$ respectively. Define $I_{2}' = I_{1}'\backslash (I_{2}\cup I_{3})$ and so on. Taking the intersection we end up with Cantor-like set. One can prove that it must be uncountable. At the same time it is empty, since the intervals cover $[0,1)$. I suppose that the desired homeomorphism just takes a point in $[0,1)$ and applies its binary walk in the case of middle thirds Cantor set to get an image. But I am puzzled, is this the desired approach? Since the cardinality argument can be applied right away...

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    $\begingroup$ terrytao.wordpress.com/2010/10/04/… Should be related. $\endgroup$
    – Ningxin
    Commented Sep 24, 2023 at 11:04
  • $\begingroup$ I'm not sure it's clear your procedure hits all the intervals? You say enumerate them, but then you must renumber them to guarantee $I_2$ on the left and $I_3$ on the right. $\endgroup$
    – M W
    Commented Sep 25, 2023 at 12:24

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By theorem of Siepiński, $[0, 1]$ can't be written as union of $2\leq \kappa\leq\aleph_0$ disjoint non-empty closed sets. In fact, the theorem of Sierpiński is stronger, it says that any compact connected metrizable space (or continuum) cannot be written as union of $2\leq \kappa\leq \aleph_0$ disjoint non-empty closed sets.

Thus if we were to write $[0, 1)$ as disjoint union of countable amount of disjoint closed intervals (or even, compact sets), then by adding $\{1\}$ to the collection, we'd also be able to write $[0, 1]$ in this way, which is impossible.

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