I have some questions concerning Alexander's horned sphere. The horned sphere is an embedding (via some continuous injection $f$) of the usual 2-sphere $S^2$ into $\Bbb R^3$ in such a way that the unbounded part of $\Bbb R^3\setminus f(S^2)$ is no longer simply connected. This is usualy used to show that the Schönflies theorem does not generalize to higher dimensions.
Questions:
- Is this embedding a topological embedding in the sense that $S^2$ is homeomorphic to $f(S^2)\subset\Bbb R^3$ given the subspace topology?
- If yes, does this means that $f(S^2)$ as a submanifold of $\Bbb R^3$ is homeomorphic to $S^2$? I ask this because I am not really sure whether a submanifold is necessarily endowed with the subspace topology w.r.t. the ambient space.
Maybe I can phrase this as a more general question.
Question: Given an $n$-dimensional (topological) manifold $M$ and an embedding $f:M\hookrightarrow\Bbb R^m$. Do we always have that $M$ is homeomorphic to $f(M)\subset \Bbb R^m$ with the subspace topology?