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?



Homeomorphisms and disjoint unions
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 $\mathbb{R}^4$.


