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I am reading "Manifolds, Vector Fields,and Differential Forms" by Gross. There we defined the notion of immersion and embedding as following:

A smooth map $F:M\to N$ is an immersion if $\mathrm{rank}_p(F) = \dim M\,\,\forall p\in M$. By an embedding, we will mean an immersion given as the inclusion map for a submanifold.

Later it was said that not every injective immersion is an embedding. I don't see why. Could you explain why this is the case and give an example?

Thanks for your help.

Schrödinger's cat
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    I didn't read this book, but usually an embedding is defined to be an immersion which is also a topological embedding, i.e a homeomorphism onto its image. Well, not every bijective continuous map is a homeomorphism. And there are indeed injective immersions which are not embeddings. – Mark Jul 17 '24 at 14:22
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    Consider $\beta\colon (-\pi, \pi) \to \mathbb{R}^2$ given by $\beta(t) = (\sin 2t, \sin t)$. Convince yourself that this is an immersion, then draw it (or have it plotted by your favorite software) and convince yourself that its image is not a manifold. (This is example 4.19 in Lee's "Introduction to Smooth Manifolds.") – Ben Steffan Jul 17 '24 at 14:23
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    This distinction was discussed many times on this site. – Moishe Kohan Jul 17 '24 at 15:01

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