# Bijection between $\mathbb{Z}\times\mathbb{Z}\times\dots$ and $\mathbb{R}$

Is there a bijection between $\mathbb{Z}\times\mathbb{Z}\times\dots$ for countably infinitely many $\mathbb{Z}$'s and $\mathbb{R}$? That is, is $\mathbb{Z}\times\mathbb{Z}\times\dots$, repeated countably infinitely many times, uncountable?

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The question after "that is" is not equivalent to the question in the first sentence; not all uncountable sets have the cardinality of $\mathbb R$. – joriki Dec 6 '12 at 9:33
What other sets do you know to be uncountable? What other sets do you know to be "bijectable" with $\mathbb{R}$? – arjafi Dec 6 '12 at 9:36

$$\mathbb{Z\times Z\times Z\times\dots = Z^N}\\\mathbb{R\times R\times R\times\dots = R^N}$$
Now we have this: $$|\mathbb R|=2^{|\mathbb N|}\leq|\mathbb Z|^{|\mathbb N|}\leq|\mathbb R|^{|\mathbb N|}=2^{|\mathbb{N\times N}|}=2^{|\mathbb N|}=|\mathbb R|$$
Can we consider any other countable set instead of $\mathbb Z$? Is the problem remained valid? Thanks Asaf. I am a disciple of you here. – Babak S. Dec 6 '12 at 9:56
Obviously we can. Any set below the cardinality of $\mathbb R$, as a matter of fact. – Asaf Karagila Dec 6 '12 at 10:01