carthesian product of two closed sets

I have a question, what is the set ? What does it contain?

I know that, the unit interval is but then

is this true?

but then we must have

I was given the following set 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?

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]).

Re: carthesian product of two closed sets

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).