# carthesian product of two closed sets

• Sep 23rd 2013, 01:10 AM
rayman
carthesian product of two closed sets
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?
• Sep 23rd 2013, 02:20 AM
chiro
Re: carthesian product of two closed sets
Hey rayman.

You are correct with your definition of [0,1] x [0,1].

As for (x,x) x in [0,1] - Consider all the points that it forms (it will form a straight line with y = x where x is in [0,1]).
• Sep 23rd 2013, 03:05 AM
rayman
Re: carthesian product of two closed sets
thank you chiro!
• Sep 23rd 2013, 05:46 AM
HallsofIvy
Re: carthesian product of two closed sets
Geometrically, [0, 1]x [0, 1] is the square in the xy-plane with verticex (0, 0), (0, 1), (1, 1), and (1, 0). {(x, x)}, x in [0, 1], is the diagonal from (0, 0) to (1, 1).