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### Writing $(a,b)$ as a disjoint union of closed intervals [duplicate]

I've been thinking about the following question: Is it possible to write $(a,b)$ as a disjoint union of closed intervals? My first guess was no, but then I figured the question might be hiding ...
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### Can the closed interval [0, 1] be expressed as the union of a sequence of disjoint closed subintervals each of length smaller than 1? [duplicate]

Can the closed interval [0, 1] be expressed as the union of a sequence of disjoint closed subintervals each of length smaller than 1? Explain. I have been trying to figure out this problem, and I do ...
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### Prove that there does not exist a sequence $I_n=[a_n,b_n]$ such that $\bigcup_{n}I_n=[0,1]$ [duplicate]

Prove that there does not exist a sequence $I_n=[a_n,b_n], n=1,2,\ldots$ of nonempty, pairwise disjoint intervals such that $\cup_{n}I_n=[0,1]$. My solution -- is it correct? My idea is that taking ...
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### Why can't a closed interval on R be written as the disjoint union of countably infinite many closed intervals? [duplicate]

As stated in the title: Why can't a closed interval $[a,b]$ on R be written as the disjoint union of countably infinite many closed intervals $[a_i,b_i]$?
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### Is $[0,1]$ a union of family of disjoint closed intervals?

According to this question, $[0,1]$ cannot be written as union of countable disjoint closed sets, is the same true about (uncountable) family of disjoint closed intervals ?
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### Can closed sets in real line be written as a union of disjoint closed intervals?

It is known that open sets in real line can be written as a countable union of disjoint open intervals. (link) I'm curious that if there is similar statements for closed sets in real line.
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### Is Every (Non-Trivial) Path Connected Space Uncountable?

I know that every non-trivial metric space with more than one point which is connected is uncountable. However, if we don't demand that the space be a metric space, we can find examples of such odd ...
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### Prove that the Cantor set cannot be expressed as the union of a countable collection of closed intervals

Prove that the Cantor set cannot be expressed as the union of a countable collection of closed intervals whereas it's complement can be expressed as the union of a countable collection of open ...
### The interval $[0,1]$ is not the disjoint countable union of closed intervals.
The following proof was suggested: suppose [0,1] was the disjoint countable union of closed intervals. Write the intervals as $[a_n,b_n]$. Start by showing the set of endpoints $a_n, b_n$ is closed. ...