Are the set of real number within the interval $[0,1]$ and the set of real number within the interval $[0,1)$ both countable?

I meet 2 questions respectively ask about the countability of the set of real number within the interval $[0,1]$ and $[0,1)$? I thought there aren't really have great difference, and are both uncountable. But I am not sure if I'm right. Do anyone has any idea? Thank you.

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    $\begingroup$ The question is unclear. Are you asking about a bijection between the two sets? $\endgroup$
    – Asaf Karagila
    Mar 9, 2013 at 14:11
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    $\begingroup$ $[0,1) \cup \{1\} = [0,1]$ $\endgroup$ Mar 9, 2013 at 14:16
  • $\begingroup$ with $[0,1)\cap\{1\}=\emptyset.$ $\endgroup$ Mar 9, 2013 at 14:17
  • $\begingroup$ They are indeed very close relatives, and in particular hsve the same cardinality. However, verifying that they do using bijections is a nice exercise in technique. $\endgroup$ Mar 9, 2013 at 14:21
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    $\begingroup$ Although this question is about $(0,1)$ and $(0,1]$, the answers can easily be adapted to produce a bijection between $[0,1]$ and $[0,1)$. $\endgroup$ Mar 9, 2013 at 14:29

1 Answer 1


Hint: If $A$ is countable, then whenever $X=A\cup\{a\}$, we have that $X$ is countable, and vice versa. If $A$ is countable then $A\setminus\{a\}$ is countable, for any $a\in A$.

Since $[0,1]=[0,1)\cup\{1\}$ either both $[0,1]$ and $[0,1)$ are countable, or both are uncountable. But going further from that, you can actually find a bijection between the two sets, which says more than their uncountability, but rather that they have the same cardinality.

To show that both are uncountable you need to prove there is no surjection from $\Bbb N$ onto $[0,1]$. Another way would be to prove that there is an injection from a set that you already know is uncountable (e.g. $\mathcal P(\Bbb N)$ or $\Bbb R$) into $[0,1)$.


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