Is $\mathbb{S}^2 \times \mathbb{R}$ homeomorphic to any subset of $\mathbb{R}^3$?

Question

I'll be working with some $2$-dimensional surfaces in $\mathbb{S}^2 \times \mathbb{R}$ soon, and this question ocurred to me. We know that $\mathbb{S}^1 \times \mathbb{R}$ is homeomorphic to $\mathbb{R}^2 \setminus \{(0,0)\}$, so in a sense an one dimensional inhabitant of the plane could ""see"" (really, walk along) curves on the cylinder, even if those curves aren't planar (of course, considering homeomorphic curves as one and the same), but is there an analogous version of this for surfaces on $\mathbb{S}^2 \times \mathbb{R}$? I have yet to build some intuition for this kind of question, so I'm sorry if it's trivial or something.

Answer 1

Just like in the case of $S^1\times \mathbb R$ you can define a homeomorphism from $S^2\times \mathbb R$ to $\mathbb R ^3 \setminus\{0\}$ via:

$$ (p,t) \mapsto e^t \, p $$

where $p$ is a point of $S^2$ when seen as the unit sphere in $\mathbb R^3$ and $t\in \mathbb R$