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Different authors use different definitions of a smooth embedding of one manifold into another: Lee's Introduction to Smooth Manifolds defines it as a smooth immersion that is also a topological embedding, but other authors define it as a smooth immersion that is injective and proper. I'd like to know if these two definitions are equivalent. Prop. 4.22 of Lee's book verifies that the second definition implies the first, but I'm having trouble with the other direction. Are all smooth immersions that are also topological embeddings proper?

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No. For instance, the inclusion map $(0,1)\to\mathbb{R}$ is a smooth immersion and a topological embedding, but it is not proper.

More generally, a topological embedding $f:X\to Y$ between locally compact Hausdorff spaces is proper iff the image $f(X)$ is closed in $Y$. (This follows immediately from the characterization of proper maps between locally compact Hausdorff spaces as closed maps with compact fibers, for instance.)

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