# injective immersions

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

$\mathbb{R}$ 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.