3D Denjoy–Riesz theorem

Question

The Denjoy–Riesz theorem states that every totally disconnected subset of $\Bbb R^2$ is the subset of a Jordan arc.

Is this true in $\Bbb R^3$? Originally I thought Antoine's necklace would be a counterexample, but now I've changed my mind. (Assuming it's not a counterexample, the resulting Jordan curve would be a very wild arc.)

Answer 1

I do not know the answer to the question, but you are correct that Antoine's necklace is not a counterexample. The necklace is contained in an Antoine's horned sphere and thus the 2d Denjoy-Riesz theorem proves the necklace lies within a Jordan curve.