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From a topological standpoint, what topological space is produced by deleting two points from an $n$-sphere $\mathbb{S}^n$ for $n \geq 3$?

  • For a 1-sphere $\mathbb{S}^1$ (i.e., a circle), deleting two points produces two lines.
  • For a 2-sphere $\mathbb{S}^2$ (a sphere), deleting two points produces a cylinder.

For both cases, $n=1$ and $n=2$, deleting two points produces a shape topologically equivalent to an $(n-1)$-sphere extruded along a perpendicular direction (e.g., twice-punctured $\mathbb{S}^2$ is equivalent to a cylinder which can be constructed from a circle in the x-y plane extruded along the z-axis).

Does this pattern extend to higher dimensions?

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Since $\mathbb{S}^n$ punctured once is homeomorphic to $\mathbb R^n$, $\mathbb{S}^n$ punctured twice is homeomorphic to $\mathbb R^n\setminus\{0\}.$

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  • $\begingroup$ Thanks. How do we know that $\mathbb{S}^n$ punctured once is homeomorphic to $\mathbb{R}^n$? $\endgroup$
    – Paul Wintz
    Commented Sep 19, 2022 at 8:23
  • $\begingroup$ @PaulWintz Stereographic projection. $\endgroup$
    – FShrike
    Commented Sep 19, 2022 at 8:27
  • $\begingroup$ Ah, right! Thanks! $\endgroup$
    – Paul Wintz
    Commented Sep 19, 2022 at 8:46

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