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It is said that topological immersion is locally 1-1 while embedding is globally 1-1, so we can say topological immersion is locally 'embedding'? (this also means except for a few points the map is 1-1, and for these points we can find a neighborhood on which the map becomes embedding?)

And embedding basically means we map a manifold onto another homeomorphic manifold in a 'surrounding'? In other words embedding is homeomorphism except that the image is now put in a bigger environment?

Examples of topological Immersion which are not embedding are $\mathbb{P}^2$ immersed in $\mathbb{R}^3$ , on the part where the immersed $\mathbb{P}^2$ intersects itself it is not 1-1 but locally 1-1.

Charlie Chang
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    How about the usual picture of the Klein bottle in $\Bbb R^3$? – Angina Seng Jul 23 '20 at 09:19
  • Yes. And if it is possible to immerse Mobius strip in $\mathbb{R}^2$ that seems to work too. I feel that I ignore the example of Klein bottle because I've not thought much of the domain the immersion of Klein bottle in $\mathbb{R}^3$? Perhaps I need to find a coordinate description of Klein bottle or find an embedding of it in $\mathbb{R}^4$ – Charlie Chang Jul 23 '20 at 09:27
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    The figure eight (lemniscate) curve is an immersion but not an embedding. See also https://math.stackexchange.com/questions/714579/bijective-immersion-is-a-diffeomorphism for more clarification. – Chrystomath Jul 23 '20 at 11:13
  • I see. Any self intersected smooth curve seems to work since it maps two points of a curve (or say (-1,1), R...) to the same point. The link also mentions that immersion is not necessarily embedding (homeomorphism btwn domain manifold and its image). I think of some say that immersion implies a function’s derivative is injective (1), but I don’t see that definition works for 8 shape, unless regarding derivatives as the pair (point p, tangent vector $v_p$)(2)? Also if so (1 and 2) then then non injective odd function like f(x)+x^3-x is not an immersion of R into R, right? – Charlie Chang Jul 23 '20 at 12:31
  • What is your definition of a topological immersion? In the smooth case it is clear. See https://math.stackexchange.com/q/1396578 – Paul Frost Jul 24 '20 at 09:01

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You are right that how the images of the parameterizations sit inside the ambient target manifold is relevant to the distinction of immersion and embedding (in any category). In my answer to Dynamics on the torus I described linear foliations on tori. On the $2$-torus, each leaf of the foliation is always injectively immersed, but each leaf is embedded iff the slope is irrational. I've also discussed a generalization of this to higher dimensions. The regularity of the immersions and embeddings are $C^\infty$ in this case, in particular they are $C^0$.

Alp Uzman
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