Isn't every connected subset of ${\mathbb R}^n$ locally a Peano space? Then the answer would be "yes".

Edit I assumed the subspace to be closed. Then it would be locally compact, hence locally Peano. I guess the main difficulty is for not closed subsets.

Isn't every connected subset of ${\mathbb R}^n$ locally a Peano space? Then the answer would be "yes".