Existence of an homeomorphic between [0,1] to X × Y I'm doing a practice exam questions and am stuck at this question:

Are there topological spaces X,Y (each with more than one point), such that [0,1] is homeomorphic to X×Y? What if we replace [0,1] with R?

I'm not even sure how to start tackle it, any help and clues will be appreciated! My head is leading me to "cut-points" area, but I'm not sure abuot it./
Thanks in advance! 

You had the right idea. $X$ and $Y$ are the image under the projections of $X\times Y$, so they must be path connected.
Now on $[0,1]$ there are many points that after removing them make this space disconnected. Can this happen with $X\times Y$, assuming both have more than one point?
(Hint: make a picture and try to connect two arbitrary points with a path.)
