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.
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5$\begingroup$ I imagine that the counterexamples to Schoenflies problem, ie. the wild spheres $S^2 \subseteq \mathbb{R}^3$ could give what you're looking for. If such a wild sphere was a submanifold, it would be locally flat and by a theorem of Morton Brown (see en.wikipedia.org/wiki/Schoenflies_problem) thus not wild at all, giving a contradiction. $\endgroup$– Piotr PstrągowskiCommented Apr 4, 2014 at 20:47
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3$\begingroup$ In fact someone already gave this example and, as I understand it, answered this question here mathoverflow.net/questions/79431/locally-flat-submanifold?rq=1 $\endgroup$– Piotr PstrągowskiCommented Apr 4, 2014 at 21:02
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1$\begingroup$ Could you include a definition of a $C^0$ submanifold? $\endgroup$– Alexandre EremenkoCommented Apr 5, 2014 at 14:01
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$\begingroup$ wiki "local flatness" $\endgroup$– PaladinCommented Apr 5, 2014 at 14:57
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