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Non-removability of Sierpinski spaces

We prove that all Sierpiński spaces in ${\mathbb{S}}^n$, $n\geq 2$, are non-removable for (quasi)conformal maps, generalizing the result of the first named author arXiv:1809.05605. More precisely, we show that for any Sierpiński space $X\subset \mathbb{S}^n$ there exists a homeomorphism $f\colon \mathbb{S}^n\to \mathbb{S}^n$, conformal in $\mathbb{S}^n\setminus X$, that maps $X$ to a set of positive measure and is not globally (quasi)conformal. This is the first class of examples of non-removable sets in higher dimensions.