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I need help finding an example that shows an immersed submanifold might have more than one topology and smooth structure with respect to which it is an immersed submanifold.

This is problem 5-15 from Lee's book on manifolds.

I thought I could use the figure eight as an example, but I'm not sure what two topologies could work.

Thanks for any help!

kumhmb
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  • there are two particular bijections of the figure eight with the real line. it inherits two different topologies from these two different bijections. – Tim kinsella Mar 27 '14 at 16:30

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As Tim Kinsella said, there are two natural bijections between $\mathbb R$ and the figure

$$\Huge 8$$

Let's say $0$ is mapped to the intersection point, then a neighborhood of $0$, $(-\epsilon,\epsilon)$, can go in either NW-SE or NE-SW direction. The limits at $\pm \infty$ will approach the intersection point from the other direction, NE-SW or NW-SE respectively.

Both maps are immersions, but the topologies they induce on figure $8$ are not identical. (They can be related by a homeomorphism, but the identity map is not a homeomorphism.)

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user127096
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