# What is the Topology of a Twice-Punctured Hypersphere (n>2)?

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 $$\mathbb R^n\setminus\{0\}.$$
• Thanks. How do we know that $\mathbb{S}^n$ punctured once is homeomorphic to $\mathbb{R}^n$? Commented Sep 19, 2022 at 8:23