TITLE: On homeomorphic subsets of $\mathbb{R}^3$ with non-homeomorphic complements
QUESTION [8 upvotes]: Let $A,B$ be two homeomorphic topological subspaces of $\mathbb{R}^3$ such that their complements $\mathbb{R}^3 - A, \mathbb{R}^3 - B$ are not homeomorphic to each other.  Must $A \cong B$ contain a homeomorphic image of the Cantor set?
(It is known that there are homeomorphic images $A,B$ of the Cantor set such that $\mathbb{R}^3 - A, \mathbb{R}^3 - B$ are not homeomorphic, see e.g. https://www.sciencedirect.com/science/article/pii/016686418690060X)

UPDATE 1:
The answer is no, as Wojowu's answer shows. This leads to
Question 2: Must $A \cong B$ contain a homeomorphic image of $\mathbb{Q}$?

UPDATE 2:
After Wojowu's answer, the interesting question remaining is
Question 3: Let $A,B$ be two closed, countable, topological subspaces of $\mathbb{R}^3$ homeomorphic to each other. Must  their complements $\mathbb{R}^3 - A, \mathbb{R}^3 - B$ be homeomorphic to each other?

REPLY [15 votes]: Let $A=\mathbb Q^3$ and $B=\{0\}\times\mathbb Q^2$. It is a classical result that they are homeomorphic (both homeomorphic to $\mathbb Q$), and their complements are not homeomorphic as $\mathbb R^3-B$ contains a subset homeomorphic to an open ball, while $\mathbb R^3-A$ doesn't (e.g. by the invariance of domain theorem).
However, since $A,B$ are countable, they don't contain a copy of the Cantor set.

The answer to the new question is also negative. Indeed let $A=\{(0,0,n)\mid n\in\mathbb N\}$ and $B=\{(0,0,1/n)\mid n\in\mathbb N\}$. Both of these are discrete and countable, so are homeomorphic. On the other hand, the complement of $A$ is a manifold, while $(0,0,0)$ in $\mathbb R^3-B$ has no Euclidean neighbourhood.