Example of non-homeomorphic spaces $X$ and $Y$ such that $X^2$ and $Y^2$ are homeomorphic

Question

Can anyone suggest an example of non-homeomorphic spaces $X$ and $Y$ such that $X \times X$ and $Y \times Y$ are homeomorphic?

Answer 1

Start with a collection of solid torii $\{T_i\}$ with each $T_{n+1}$ embedded in $T_n$ as follows

Then the complement of the nested intersection, namely, $\Bbb S^3 \setminus \bigcap T_i$ is called the Whitehead manifold, which we will denote by $W$. $W$ is not homeomorphic to $\Bbb R^3$ as it is not simply connected at infinity (which is surprising, as $W$ is contractible since each $\Bbb S^3\setminus T_n$ is nullhomotopic in $\Bbb S^3 \setminus T_{n+1}$)

In Glimm, Two cartesian products which are euclidean spaces, Bull. Soc. Math. France $88 (1961), 131-135$, available here, it is proved that $W \times W \cong \Bbb R^6 \cong \Bbb R^3 \times \Bbb R^3$.

Thus, $X = W$ and $Y = \Bbb R^3$ provides an example of such a space.