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In Spivak's book on differential geometry he defines a topological immersion $f$ as "$f$ is a continuous function that is locally one-one". In my limited experience with the category $\mathsf{Top}$, this seems like the wrong definition, and I was wondering if anyone here agreed.

Wouldn't the definition consistent with the $\mathsf{SM}$ definition be "$f$ is a local homeomorphism onto its image"? It seems strange to think that for instance, every injective map into an indiscrete space falls under the heading of immersion.

Eric Auld
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    A smooth immersion is not a local diffeomorphism: immersion here just means that the induced map on tangent spaces is injective, while "local diffeomorphism" would require that the induced map be bijective. – Santiago Canez Aug 14 '15 at 02:16
  • This is relevant: http://math.stackexchange.com/questions/1023163/immersion-locally-injective – Santiago Canez Aug 14 '15 at 02:21
  • @SantiagoCanez Excuse me, the map to the image is a local diffeomorphism – Eric Auld Aug 14 '15 at 03:06
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    A one-one continous function may NOT be a local homeomorphism onto its image.

    Consider $f$ a continuous injetive function from $[0,5)$ into $\mathbb{R}^2$, such that $$ f(x)= \begin{cases} (x,0), &\textrm{ if } x\in[0,2) \ (2,x-2), &\textrm{ if } x\in [2,3) \ (5-x,1), &\textrm{ if } x\in [3,4) \ (1,5-x), &\textrm{ if } x\in [4,5) \end{cases} $$ Pick the point $p=1$ in the domain. Its image is $f(p)=(1,0)$. There is not any neighbourhood $V$ of $p$ and any neighbourhood $W$ of $f(p)$, such that $f$ is a homeomorphism between $V$ and $W$.

     – Ramiro Aug 14 '15 at 04:30
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    This example can be easily adapted to be a smooth immersion that is not a local diffeomorphism. – Ramiro Aug 14 '15 at 04:35
  • @RamiroGuerreiro However, as Mike alludes to below, a sufficient condition for this to be true is locally compact domain and Hausdorff codomain. – Eric Auld Aug 14 '15 at 05:54
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    @EricAuld Please, note that the real interval $[0,5)$ is locally compact and $\mathbb{R}^2$ is Hausdorff. The function $f$ (I presented above) is a local embedding, in the sense that, for every $p \in [0,5)$, there is $V$ neighbourhood of $p$ such that $f$ is a homeomorphism between $V$ and $f(V)$. BUT, if $p=1$, then $f(V)$ is NOT open in the image $f([0,5))$ (with the topology induced by $\mathbb{R}^2$). So $f$ is NOT a local homeomorphism. – Ramiro Aug 14 '15 at 12:26
  • @RamiroGuerreiro Ah, this is helpful because I didn't know that requirement on a local homeomorphism. What would you call a function with the first property you mention? Perhaps that is the proper definition of "topological immersion". – Eric Auld Aug 14 '15 at 17:02
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    Local (topological) embedding: a function $f$ such that for every $p$ in the domain there is $V$ neighbourhood of $p$ such that $f$ is a homeomorphism between $V$ and $f(V)$. (Since $f(V)$ is not required to be open in the image of $f$, a local embedding may not be a local homeomorphism). – Ramiro Aug 14 '15 at 17:14
  • @RamiroGuerreiro Thanks, I think that fully answers my question in the OP. I think an immersion should be a "local embedding", so I would edit Spivak's definition. But it works for locally compact --> Hausdorff, but I will be careful to appreciate the difference in a "local homeomorphism" – Eric Auld Aug 14 '15 at 17:40
  • @Ramiro, Eric Auld, what is a local diffeomorphism onto image exactly please? I know what local diffeomorphisms, local homeomorphisms onto image and local homeomorphisms are. The issue with copying the definition of local homeomorphism onto image to local diffeomorphism onto image has the problem of the image possibly not being a submanifold or manifold while there is no such issue for local homeomorphism onto image since image can always be made into a subspace. –  Jul 22 '19 at 05:55

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Surely you mean "$f$ is locally (in the domain) an embedding" rather than homeomorphism. In any case, the goodness or badness of a definition depends precisely on what he wants to use it for. What does Spivak want to use this notion of 'topological immersion' for? Note that for topological manifolds, 'local embedding' and 'locally injective' are the same! I doubt he had non-Hausdorff spaces in mind here.

This notion of immersion has historical significance, in the case that the domain and codomain are both manifolds. See for instance Hirsch, "On piecewise linear immersions".

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    Yes, excuse me, the map to the image is a local homeomorphism. – Eric Auld Aug 14 '15 at 03:05
  • @EricAuld what is a local diffeomorphism onto image exactly please? I know what local diffeomorphisms, local homeomorphisms onto image and local homeomorphisms are. The issue with copying the definition of local homeomorphism onto image to local diffeomorphism onto image has the problem of the image possibly not being a submanifold or manifold while there is no such issue for local homeomorphism onto image since image can always be made into a subspace. –  Jul 22 '19 at 05:55