# Is product of countably many copies of $\{0,1\}$ uncountable?

Let $X$ denote the two point set $\{0,1\}$ and let $X_j=\{0,1\}\forall j=1,2\dots$ let $Y=\Pi_{j=1}^{\infty}X_j$, I need to determine whether each of the following are true or false:

1. $Y$ is countable

2. $|Y|$=|[0,1]|

3. $\bigcup_{n=1}^{\infty}\Pi_{j=1}^{n}X_j$ is uncountable

4. $Y$ is uncountable.

I guess $Y$ is uncountable (4), but I can not prove it.

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Do you have any ideas about any of the four? –  Brian M. Scott Jun 13 '12 at 10:42
We have to show any finction $g:\mathbb{N}\rightarrow Y$ is not surjective. –  La Belle Noiseuse Jun 13 '12 at 10:45
Is #3 is a typo? You probably meant $n$ for the second infinity? –  rschwieb Jun 13 '12 at 11:02
Yes, I guess so. –  La Belle Noiseuse Jun 13 '12 at 11:04
@mex If you start probing why you feel justified in guessing your guess, more of your guesses will turn into solutions! If you include more of your thoughts about your guess, we can help you do this. –  rschwieb Jun 13 '12 at 11:53

2. Every real in $[0,1]$ has a binary decimal expansion $0.b_1b_2b_3\ldots$
Every $a \in Y$ is an infinite series of zeros and ones. Think of any $b \in P(\mathbb N)$ and try to find the connection between this two.