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?

(The analogous statement in 2 dimensions is true, by Richards' classification of non-compact surfaces: https://www.ams.org/journals/tran/1963-106-02/S0002-9947-1963-0143186-0/S0002-9947-1963-0143186-0.pdf.)

nonhomeomorphic subspaces of $\mathbb R^3$, does this imply that $\mathbb R^3 \setminus A$ and $\mathbb R^3 \setminus B$ are not homeomorphic? $\endgroup$1more comment