For a given subspace $X \subset ℝ^n$ there is a homotopy equivalence $X \times \{0\} \simeq X \times I$ (subspaces of $ℝ^{n+1}$), where $I$ is the unit interval. However, the complements in $ℝ^{n+1}$ are not necessarily homotopic.
The complements should be homotopic for nice spaces $X$, in particular when $X$ is closed (hopefully). Unfortunately I fail to find a reference for this, could someone please point me in the right direction?
Example where the complements differ:
Let $N = ℝ \setminus (\{\frac{1}{2^n}|n\in \mathbb{N}\}\cup \{0\}$) then $\pi_1(ℝ^2 \setminus (N\times 0))$ is larger than $\pi_1(ℝ^2 \setminus (N\times I))$ because the former contains loops with infinitely many circles, the later doesn't.
This is a special case of a much broader question asked a year ago. The older question is wrong (does not sufficiently restrict the class of allowed sets $X$) and has no (to me) useful answer.