TITLE: Homeomorphisms and disjoint unions
QUESTION [32 upvotes]: Let $X$ and $Y$ be compact subsets of $\mathbb{R}^n$.  Assume that $X \sqcup X \cong Y \sqcup Y$ (here $X \sqcup X$ is the disjoint union of two copies of $X$, considered as a topological space, and similarly for $Y \sqcup Y$).  Then I'm pretty sure that we must have $X \cong Y$.  This clearly holds if $X$ and $Y$ are connected, but I can't seem to prove it in general.  Can anyone help me?

REPLY [35 votes]: The result you want is false.  Counterexamples are given in
Yamamoto, Shuji and Yamashita, Atsushi,
A counterexample related to topological sums. 
Proc. Amer. Math. Soc. 134 (2006), no. 12, 3715–3719.
These counterexamples are compact subsets of $\mathbb{R}^4$.