I read from a text book that unit circle ($\mathbb{S}$) is not diffeomorphic to the real line. This result is intuitive since we cannot construct a smooth function from $\mathbb{S}$ to $\mathbb{R}$ such that it is onto (e.g., $f(\rho)=(\cos(\rho),\sin(\rho))$ defined from $\mathbb{R}$ to $\mathbb{S}$ is not onto). How can I prove this result?
Incidentally, I am not familiar with advanced topics in topology. Thank you, in advance, for your response!
The two sets, the unit circle and the real line are not even homeomorphic.
Note that removing a point from the real line makes it disconnected but removing a point form the unit circle keeps it connected.