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