Let $K\subset \mathbb{R^2}$ be the Koch fractal. We have by construction the injective and continuous map $S^1\rightarrow K$ that takes a finite arc length to infinity.
Do we have the inverse map $K\rightarrow S^1$, and is there any ambiguities with the Jordan curve theorem?
EDIT: This question came to my mind when I read Wikipedia about wild arcs: "a wild arc is an embedding of the unit interval into 3-dimensional space not equivalent to the usual one in the sense that there does not exist an ambient isotopy taking the arc to a straight line segment."
Yes, try to prove the following fact: Any continuous bijective map from a compact space into a Hausdorff space is a homeomorphism.