Is $S^1\times S^2$ isomorphic to $\mathbb{R}P^3$?

Question

My question here is simple. Is it the case that $S^1\times S^2\cong \mathbb{R}P^3$?

This seems like this should be true for a couple of reasons. First, $\mathbb{R}P^3\cong SO(3)$ and every rotation is uniquely determined by an axis, and an angle about that axis, i.e. an element of $S^2$ and an element of $S^1$. It is also the case that $SO(3)/SO(2)\cong SO(3)/S^1\cong S^2$. However, I can't seem to find a way to show that the two spaces are topologically isomorphic.

If they're not isomorphic, then how exactly are $S^1\times S^2$ and $\mathbb{R}P^3$ related?

Answer 1

As pointed out in Tsemo Aristide's answer, the two spaces are not homeomorphic. However, what you have found is the Hopf fibration.

Answer 2

The fundamental group of $S^1\times S^2$ is $\mathbb{Z}$ and the fundamental group of $\mathbb{R}P^3$ is $\mathbb{Z}/2$, they are not homeomorphic.