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Is there an injective immersion between smooth manifolds that is no homeomorphism onto its image? With smooth I mean $C^\infty$-manifolds and of course also the immersion should be $C^\infty$.

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There is an injective immersion of $\mathbb{R}$ into the plane, whose image is the figure 8. Clearly it it not an homeomorphism to its image (since this is not a manifold).

See also: http://en.wikipedia.org/wiki/Immersed_submanifold#Immersed_submanifolds

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Thanks. That was easy. I wonder why I didn't figure that out by myself. –  principal-ideal-domain Feb 27 '14 at 13:47
Often counterexamples are easy once you "see" it. Coming up with them in the first place is the hard part. The counterexample I stated is the standard one. –  Thomas Rot Feb 27 '14 at 13:49
But sometimes it's hard to verify that your example is really a counterexample. But of course not in that case. –  principal-ideal-domain Feb 27 '14 at 13:51

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