I have a question, what is the set $\displaystyle [0,1]\times[0,1]$? What does it contain?

I know that, the unit interval is $\displaystyle [0,1]:=\{x\in\mathbb{R}|0\lex\le 1$ but then

$\displaystyle [0,1]\times [0,1]=\{(x,y)|0\le x\le 1,0\le y\le 1|x,y\in\mathbb{R}\}$ is this true?

but then we must have $\displaystyle [0,1]\times [0,1]\subset \mathbb{R}^2$

I was given the following set $\displaystyle A=\{(x,x)|x\in[0,1]\}$ what is it and what does it contain? is it just a set of open intervals of the real line?

or is it a diagonal of the unit square?