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if my function is an immersion and say it is defined on a path connected open of an euclidian space.

immersions are locally injective. if we add the path connectedness could I assert that my immersion is injective? If not I would be grateful to you for giving me a counterexample.

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Think of a periodic parametrization of the figure eight by the reals, for example. –  t.b. May 15 '11 at 2:31

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$\mathbb{R}$ is path connected, and $f:\mathbb{R}\rightarrow\mathbb{R}^2$ defined by $f(t)=(\cos(t),\sin(t))$ is an immersion because $$d_tf=[-\sin(t)\quad\cos(t)]$$ is never of rank 0, but it is not injective.

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Of course, I agree, but the downside of this example is that the image still is an embedded submanifold, so it fails to exhibit one important feature of immersions, namely that you get an embedded manifold locally, but not globally. –  t.b. May 15 '11 at 2:36
    
I can see where my confusion comes from: usually when we have a property P that is locally verified in a connected space and if we prove that P is true on an open and closed part of a space then P is valid on the whole space –  El Moro May 15 '11 at 2:37

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