So you are saying that supposing we had a set S with points/elements (rational numbers) such as {1.1, 1.2, 1.3, ...} - gotten from a real number line, we could pick any point, say 1.1 and scale/magnify it down by 0.1, and correspondingly we have points which is also a set, given below as

X = { 0.11, 0.22, 0.33,...., 1.1}

X has cardinality of 10

Which means that every point in S when scaled/magnified by a non-zero factor gives a new set of points ( e.g X for 1.1), and every point in this new set also gives it's own corresponding new set when also magnified, and so those the process continues to ad infinitum. The self-similarity here is that points are contained in a point when magnified. However, a single point has a topological dimension of zero which is equivalent to the point's fractal dimension but the fractal dimension (box-counting dimension) of any magnified point, say 1.1 in S above is given by

D = log(N)/log(1/e)

D = fractal/box-counting dimension
N = number of points in the magnified point (1.1)
e = scale/magnification factor.

In our scenario as seen above

N = 10 (cardinality of set x)
e = 0.1
Therefore D = 1

This seems to me like a fractal.