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?
 A: 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 $
$A_2 = B_2  \cup B_3$
A: Yes. In fact, you can find an uncountable collection of uncountable dense sets.
The article “Partitioning the Real Line into an Uncountable Collection of Everywhere Uncountably Dense Sets” by Seth Zimmerman and Chungwu Ho in The American Mathematical Monthly (vol. 126, no. 9, November 2019, p.825) will be of interest.
A:  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?
What I meant was
We can generalize the process as 
$$A_1=\lim_{n\to\infty}\bigcup_{i=1}^{n}[(2i-2)/2n,(2i-1)/2n)]$$
$$A_2=\lim_{n\to\infty}\bigcup_{i=1}^{n}[(2i-1)/2n,2i/2n]$$
