Covering spaces and fibre bundles, using the of $SU(2)$ and $SO(3)$

Question

I understand there are many questions on here that show give an explicit map to show that $SU(2)$ is a double cover of $SO(3)$ (seevia quaternions).

I am trying and use the fact that $SU(2)$ is a double cover of $SO(3)$ to write $SU(2)$ as some fibre bundle. But I seem to encounter some contradictions. I want to know what is incorrect. Here are my ideas:

(1) Since the Lie algebras of $SO(3)$ and $SU(2)$ are isomorphic. They have isomorphic connected components.

(2) Covering spaces can always be thought of as discrete fibre bundles over the base space.

(3) But $SU(2)$ is simply connected.

I think (2) is wrong.

(4) If two is wrong what is the correct way to think about covering spaces. As far as I understand $SU(2)$ has one connected component, and $SO(3)$ has two connected components. So it seems like maybe $SO(3)$ should be a double cover of $SU(2)$?

Answer 1

(1) Is wrong. The kernel of $SU(2)\to SO(3)$ is $\{\pm I\}$. The element $-I$ is central in $SU(2)$, but $SO(3)$ has trivial center, so they cannot be isomorphic. (Both $SU(2)$ and $SO(3)$ are connected.)

(2) True.

(3) True.

(4) It's wrong to say $SO(3)$ has two connected component.

It's true to say $SO(3)$ is the identity component of $O(3)$, the unique other component being comprised of reflections (and "improper reflections").