I've been looking over problems in algebraic topology and figuring out questions dealing with homotopy equivalent spaces. I noticed the following problem, but can't formally verify my answer.
Determine whether or not $X = \mathbb{R}^3 \setminus \{p\}$ and $Y = \mathbb{R}^3 \setminus l$ are homotopy equivalent, where $\{p\}$ is a point and $l$ is a line.
Intuitively, it seems as though they are not, but as mentioned, I'm having a difficult time showing this. Would I be able to somehow show they are not using their fundamental groups? So far, I know that $\pi_1(X) = \{e\}$.
Can anyone help?
Thank you!