Show then the set R (the reals) with the Euclidean distance is homeomorphic to to the set (0,1) with the same distance, are they isometric?
My idea was some sort of scaled down tan function, is the correct? Can anyone be more precise?
Show then the set R (the reals) with the Euclidean distance is homeomorphic to to the set (0,1) with the same distance, are they isometric?
My idea was some sort of scaled down tan function, is the correct? Can anyone be more precise?
Yes, that is the right idea. To be more precise, the invertible continuous function $\displaystyle f(x) = \tan\bigl((x-\tfrac12)\pi\bigr)$ maps the open unit interval onto the whole real line.
Of course there can never be an isometric map between these two spaces, because the line contains points whose distance apart is greater than 1, whereas the unit interval doesn't.