Exercise 586Topology › Topology of ℝⁿL1Added 3 Nov 2026

Boundary of a ball

#boundary#sphere#ball

Statement

Compute the boundary of the closed ball $\overline B(a,r)\subseteq\mathbb{R}^n$ .

Hint

Boundary = closure $\setminus$ interior. The closed ball is its own closure; its interior is the open ball.

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Closure. $\overline B(a,r)$ is closed, so its closure is itself.

Interior. $\mathring{\overline B}=B(a,r)$ : any boundary point fails to have a ball inside the closed ball.

Boundary. $\overline{B(a,r)}\setminus B(a,r)=\{x:\|x-a\|=r\}$ .

Conclusion. The boundary is the sphere $S(a,r)=\{x:\|x-a\|=r\}$ .