Are topological 1-manifolds embedded in 2-manifolds always locally flat?

Question

I would like to know whether the following is true:

Let $M$ be a topological 2-manifold (without boundary), and let $i: [0,1] \to M$ be a continuous embedding. Then $\mathrm{Im}(i)$ is locally flat in $M$.

This question has been asked before, but I could not obtain the statement from the reference given in the comments. As pointed out there, the counterexamples to local flatness that are commonly given take place one dimension higher.

Answer 1

The statement you want does follow immediately from the theorem quoted in the second comment of the link you provided:

• ... if a subset of $\mathbb R^2$ is abstractly triangulable, then there is a homeomorphism $f$ of the plane taking it to a polyhedron of $\mathbb R^2$, which is of course locally flat.

Simply use that $i[0,1]$ is homeomorphic to $[0,1]$ and hence abstractly triangulable.