Initially on [0,1], all elements should have a one-one relation with ℝ. I thought that the deleted open middle-third intervals of the Cantor Set when deleted lose out on their images in ℝ so essentially there will now be some images on ℝ which have no pre-images on the Cantor Set and thus there won't be a one-one relation with ℝ. But I am not very sure of this and even if I'm right a better explanation by someone else would really help a lot
Each element of the Cantor set C, can be expressed as a decimal base
three without any 1's. Take that expression and change every 2 to
a 1 and read the result as a decimal base two. Thus a bijection
between C and [0,1].