It is shown that every perfect $G_δ$ subset of the real line can be parted onto $n$ homeomorphic parts; every open subset of $\mathbb{R}^m$ can be parted onto n homeomorphic parts provided $n ≥ 2m + 2$;  no compact convex subset of $\mathbb{R}^m$ can be parted onto two congruent parts.

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