Consider a spherical surface S. We can construct a mapping from the point of S to the points in a finite and limited subset of R^2 (say, the square [0, 1] X [0, 1]). One way to do that is to map each point of "latitude" ρ and "longitude" θ (in degrees), to the point ((ρ+90)/180, θ/360).
Now, let's consider the rectangle [0, .5] x [0, 1]. There is obviously a 1:1 mapping from this to the original square (just halve or double one of the coordinates to change from one to the other). In an intuitive sense, there are "as many" points in half the square as in the whole square, a bit like there are as many natural numbers as there are even numbers. (This is one definition of infinite set, if I remember correctly).
Now this means that we can remap all the points from half the square to a complete sphere, and the points from the other half square to a "new" complete sphere, in fact mapping points from the original sphere S to two new identical spheres.
Is this reasoning correct, or am I missing anything (probably something obvious...)?