Non-removability of Sierpiński spaces

<p>We prove that all Sierpiński spaces in <inline-formula content-type="math/mathml">
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-struck upper S Superscript n">
  <mml:semantics>
    <mml:msup>
      <mml:mrow class="MJX-TeXAtom-ORD">
        <mml:mrow class="MJX-TeXAtom-ORD">
          <mml:mi mathvariant="double-struck">S</mml:mi>
        </mml:mrow>
      </mml:mrow>
      <mml:mi>n</mml:mi>
    </mml:msup>
    <mml:annotation encoding="application/x-tex">{\mathbb {S}}^n</mml:annotation>
  </mml:semantics>
</mml:math>
</inline-formula>, <inline-formula content-type="math/mathml">
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="n greater-than-or-equal-to 2">
  <mml:semantics>
    <mml:mrow>
      <mml:mi>n</mml:mi>
      <mml:mo>≥<!-- ≥ --></mml:mo>
      <mml:mn>2</mml:mn>
    </mml:mrow>
    <mml:annotation encoding="application/x-tex">n\geq 2</mml:annotation>
  </mml:semantics>
</mml:math>
</inline-formula>, are non-removable for (quasi)conformal maps, generalizing the result of the first named author [<italic>Non-removability of Sierpiński carpets</italic>,  2018]. More precisely, we show that for any Sierpiński space <inline-formula content-type="math/mathml">
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper X subset-of double-struck upper S Superscript n">
  <mml:semantics>
    <mml:mrow>
      <mml:mi>X</mml:mi>
      <mml:mo>⊂<!-- ⊂ --></mml:mo>
      <mml:msup>
        <mml:mrow class="MJX-TeXAtom-ORD">
          <mml:mi mathvariant="double-struck">S</mml:mi>
        </mml:mrow>
        <mml:mi>n</mml:mi>
      </mml:msup>
    </mml:mrow>
    <mml:annotation encoding="application/x-tex">X\subset \mathbb {S}^n</mml:annotation>
  </mml:semantics>
</mml:math>
</inline-formula> there exists a homeomorphism <inline-formula content-type="math/mathml">
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="f colon double-struck upper S Superscript n Baseline right-arrow double-struck upper S Superscript n">
  <mml:semantics>
    <mml:mrow>
      <mml:mi>f</mml:mi>
      <mml:mo>:<!-- : --></mml:mo>
      <mml:msup>
        <mml:mrow class="MJX-TeXAtom-ORD">
          <mml:mi mathvariant="double-struck">S</mml:mi>
        </mml:mrow>
        <mml:mi>n</mml:mi>
      </mml:msup>
      <mml:mo stretchy="false">→<!-- → --></mml:mo>
      <mml:msup>
        <mml:mrow class="MJX-TeXAtom-ORD">
          <mml:mi mathvariant="double-struck">S</mml:mi>
        </mml:mrow>
        <mml:mi>n</mml:mi>
      </mml:msup>
    </mml:mrow>
    <mml:annotation encoding="application/x-tex">f\colon \mathbb {S}^n\to \mathbb {S}^n</mml:annotation>
  </mml:semantics>
</mml:math>
</inline-formula>, conformal in <inline-formula content-type="math/mathml">
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-struck upper S Superscript n Baseline minus upper X">
  <mml:semantics>
    <mml:mrow>
      <mml:msup>
        <mml:mrow class="MJX-TeXAtom-ORD">
          <mml:mi mathvariant="double-struck">S</mml:mi>
        </mml:mrow>
        <mml:mi>n</mml:mi>
      </mml:msup>
      <mml:mo class="MJX-variant">∖<!-- ∖ --></mml:mo>
      <mml:mi>X</mml:mi>
    </mml:mrow>
    <mml:annotation encoding="application/x-tex">\mathbb {S}^n\setminus X</mml:annotation>
  </mml:semantics>
</mml:math>
</inline-formula>, that maps <inline-formula content-type="math/mathml">
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper X">
  <mml:semantics>
    <mml:mi>X</mml:mi>
    <mml:annotation encoding="application/x-tex">X</mml:annotation>
  </mml:semantics>
</mml:math>
</inline-formula> 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.</p>