I was introduced to space filling curves recently and it occurred to me that it seems you could use them to parametrize any surface in $\mathbb R^3$ using only one real parameter, which is as far as I know not possible.
From what I understand, a space filling curve such as the Hilbert curve is a homoemorphism between the real line and a subset of the plane. Since a composition of homeomorphisms is a homeomorphism, we could compose the surface patch(es) of say a sphere with the Hilbert curve to get a homeomorphism between the real line and the sphere. But this is impossible since the sphere is a 2-manifold. Any guidance as to where I go wrong would be much appreciated!
I have to say I was only introduced to the Hilbert curve informally so this is probably the root of my confusion.
Generally: any injective map from a compact space to a Hausdorff space is always a homeomorphism from the domain to the range. Hence, any injective map from a close interval to a plane will look exactly like a curve, 1-dimensional and barely capable of filling anything. Space filling curve is not injective.