Schönflies theorem

《Mathematical Physics: Classical Mechanics》 · 第 19 页 · Chapter 2，Theorem 2.9（完整分段线性三维版本）

Let \(e:S^2\to S^3\) be any piecewise linear embedding. Then \(S^3-eS^2\) has two components, the closure of each of which is a piecewise linear ball.

《Chromatic Homotopy Theory》 · 第 276–277 页 · Section 9.2.3，Definition 9.2.7 的不可约条件与 Theorem 9.2.10（Alexander’s Theorem）

The manifold \(M\) is irreducible if every sphere \(S\subset\operatorname{int}(M)\) bounds a ball. It is important to recall here that the sphere must be a smooth surface. […] The space \(\mathbb R^3\) is irreducible.