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I am attempting the following exercise:

Let $M$ be a connected manifold. Show that there is a smooth proper map $f:M\to[0,\infty)\subset\mathbb{R}$.

Surely I can just take $f$ to be the zero map; that is, $f(x):=0$ for all $x\in M$.

This is proper since the preimage of every compact set will be either $\varnothing$ or $\{0\}$, both of which are compact. It is also smooth.

Am I missing something here? I have not used the assumption that $M$ is connected.

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  • $\begingroup$ The preimage of a compact set containing $0$ is all of $M$... $\endgroup$ – studiosus Apr 10 at 12:04
  • $\begingroup$ It is correct and you don’t need to use that M is connected $\endgroup$ – Federico Fallucca Apr 10 at 12:09
  • $\begingroup$ @studiosus I do not know that $M$ is compact, so does the zero map not work here? $\endgroup$ – rbird Apr 10 at 12:21
  • $\begingroup$ @rbird Indeed, the zero map does not work when $M$ is not compact. $\endgroup$ – studiosus Apr 10 at 12:39

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