My question arises from Chapter 21 (page 541-542, example (c)) of Lee's book, Introduction to Smooth Manifolds, 2nd edition. I find in Lee's book ''Introduction to Smooth Manifolds'' this example. He says that the quotient map is f : $\Bbb C$ $\rightarrow$ [0,+$\infty$). I understood this point. But i can't comprehend how he comes to a conclusion that the orbit space is not a manifold. And then, he says that if we delete the origin, which means that we will have $\Bbb C\setminus\{0\}$ instead of $\Bbb C$, the orbit space is a manifold. Why? Any hints/help? Thanks in advance.
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A manifold locally resembles Euclidean space near each point. Bear in mind that $\ln x$ maps $(0,,\infty)$ to $\Bbb R$, but we can't do anything analogous with $[0,,\infty)$. – J.G. Mar 23 '19 at 16:08
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Thank you very much! – Ioannis Vousnakis Mar 23 '19 at 16:15