# Cantor set and expressing $[0,1)$ as a countable union of disjoint closed intervals

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...

• terrytao.wordpress.com/2010/10/04/… Should be related. Commented Sep 24, 2023 at 11:04
• 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.
– M W
Commented Sep 25, 2023 at 12:24

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.