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I am solving the following problem (the motivation is that the inclusion of the open unit disk in the plane is not proper):

Problem: Let $M$ be a connected, non-compact topological manifold without a boundary, and $N$ be a submanifold of $M$ such that $M$ is homeomorphic to $N$ and $N\subsetneqq M$ (i.e., $N$ is a proper subset of $M$). Then the inclusion map $i\colon N\hookrightarrow M$ is not a proper map.

My Thought: $M$ is homeomorphic to $N$ implies $\dim M=\dim N$ and $\partial M=\varnothing =\partial N$. I want to show that the image of the injective map $i\colon N\hookrightarrow M$ is an open set of $M$. Pre and post-composing $i\colon N\hookrightarrow M$ with Euclidean charts, and then using invariance of domain, we can say that $i(U)\subseteq_\text{open}M$, if $U$ is the domain of a Euclidean chart. Now, covering $N$ by such open sets, we are done, i.e., $N$ is an open subset of $M$. Now, a proper map between two topological manifolds is a closed map; see this answer. Thus $i\colon N\hookrightarrow M$ can NOT be a proper map as $M$ is connected and $\varnothing \neq N\neq M$ .

Is my thought correct??

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1 Answers1

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Your argument looks correct to me. Note that depending on the definition of submanifolds, you may need to add the assumption that $N$ does not have boundary.

In fact an inclusion with locally compact Hausdorff target is a proper map iff it has closed image.

ronno
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