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An embedded submanifold is an immersed submanifold for which the inclusion map is a topological embedding. A properly embedded submaniold is one which is embedded and the inclusion map is proper. There are many classical examples of one-to-one immersions which are not emeddings e.g. a line of irrational slope on the 2-torus. The assiciated inclusion map in this case is obviously not proper, but I am having trouble thinking of an embedded submanifold which is not properly embedded. There are examples when the ambient manifold is not Hausdorff, for instance, but can someone think of an example when both manifolds are well behaved?

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Think of the embedding $\mathbb R^n\to\mathbb S^n$, which image is $\mathbb S^n\setminus \{N\}$. Then, for any compact neighborhood $C$ of $N$ in $\mathbb S^n$, the inverse image of $C$ is not a compact space.

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