Space filling curves

Question

The classic Hahn-Mazurkiewicz theorem has the following consequence: Let $X$ be a compact, connected topological manifold. Then there is a continuous surjective map $f: [0,1] \rightarrow X$.

It is also true that such a $f$ cannot be injective unless $X=[0,1]$ (since then $f$ would be a homeomorphism). I was wondering if some weaker notions of this is possible, where injectivity holds (this would be very useful for my research). Specifically, suppose we have a compact, connected Riemannian manifold $(M,g)$, without any boundary. Let $d$ be the metric induced on $M$ by $g$. Then I want to know the following:

For every $\epsilon > 0$, does there exist a map $f_{\epsilon}: [0,1] \rightarrow M$ that is continuous and injective, such that for any $x \in M$ there exists $y \in [0,1]$ such that $d(x,f_{\epsilon}(y)) < \epsilon$?

Does anything change if $M$ is non-compact?

Answer 1

I would call such $f_\epsilon$ coarsely $\epsilon$-dense, or more simply $\epsilon$-dense. These exist in the compact case as follows. Tile $M$ by $n$-balls (these may overlap, but only finitely). Each ball is homeomorphic to $[0, 1]^n$, so admits an $\epsilon/2$-dense smooth Hilbert arc. A small perturbation of these makes them pairwise disjoint. Choose an ordering of the $n$-balls. Now connect the Hilbert arcs together, in order. Tubular neighbourhoods allow us to do this smoothly and injectively.

In the complete, non-compact case this is not possible. This is because the continuous image of a compact set is compact, and thus contained in a metric ball.