The circle group acts on complex plane by complex multiplication.

Question

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.