Can two uncountable disjoint sets be dense in $[0,1]$? I am not sure whether this question was already asked? Let me know if so.
Suppose $A_1$ and $A_2$ are uncountable disjoint subsets of $A$. Can $A_1$ and $A_2$ be dense (meaning "closely approximate all points") in $[0,1]$? Is it possible to give an elementary example? I haven't studied real analysis in college yet.
 Edit
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What about...
$$\require{enclose} \enclose{horizontalstrike}{A_1=\lim_{n\to\infty}\bigcup_{i=1}^{ \lceil n/2 \rceil}[0,2i/n]}$$
$$\require{enclose} \enclose{horizontalstrike}{A_2=\lim_{n\to\infty}\bigcup_{i=1}^{\lceil n/2 \rceil}[2i/n,(2i+1)/n]}$$
Would $A_1$ and $A_2$ be uncountable?
Second Edit:
Here's  what I really meant
$$A_1=\lim_{n\to\infty}\bigcup_{i=1}^{n}[(2i-2)/2n,(2i-1)/2n)]$$
$$A_2=\lim_{n\to\infty}\bigcup_{i=1}^{n}[(2i-1)/2n,2i/2n]$$
Are $A_1$ and $A_2$ uncountable?

Let 
$B_1 = \left (0, \frac{1}{2} \right)  \cap \mathbb Q$
$B_2 = \left( 0, \frac{1}{2} \right) - \mathbb Q$
$B_3 = \left(\frac{1}{2}, 1 \right)  \cap \mathbb Q$
$B_4 = \left(\frac{1}{2}, 1 \right)  - \mathbb Q$
Then
$A_1 = B_1  \cup B_4 $
$A_2 = B_2  \cup B_3$
