A space-filling plane curve is a continuous surjective function from the unit interval $[0,1]$ to the unit square. Netto's theorem gives that continuous bijections preserve dimension, so a space-filling curve cannot be injective. In fact, for any given space-filling curve (e.g. the Peano or Hilbert curve), there are many points on the square whose preimage consists of multiple points in the unit interval.
For a given plane-filling curve (e.g. Peano or Hilbert), are there two specific computable real numbers in the unit interval that are known to map to the same point in the square? Do we have any characterization of the number of points in the square whose preimage contains multiple elements? Are those preimages always finite, or do some consist of infinitely many numbers in the unit interval?
The Hilbert curve definitely has the property that it maps certain pairs of computable numbers to the same point in the plane. Moreover, to see this we only need to consider rather few properties of it. I've explained this over at Mathoverflow:
Is the Hilbert curve bijective over computable numbers?@MathOverflow
It is not immediately obvious to me whether the argument extends to all computable surjections $\tau : [0,1] \to [0,1]^2$. Basically, any space-filling curve that "aims to be as injective as possible" will (ironically) fail to be injective on the computable reals. But maybe something which is highly non-injective on the non-computable reals could achieve being on the computable reals?