At first I thought that if a subspace of $\mathbb{R}^n$ is homeomorphic to a manifold, then it is a $C^0$ submanifold of $\mathbb{R}^n$. But I found an asterisked exercise in the book Differential Topology by Morris Hirsch that said ``if a subset of $\mathbb{R}^2$ is homeomorphic to $S^1$ then it is a $C^0$ submanifold of $\mathbb{R}^2$'' which requires Schoenflies' Theorem to prove (given as a hint). As this special case requires that much machinery, I think in general it is not true.