Exercise 587Topology › Topology of ℝⁿL1Added 3 Nov 2026

Interior, closure, boundary

#interior#closure#boundary

Statement

Define the interior, closure and boundary of $A\subseteq\mathbb{R}^n$ , and compute them for $A=B(a,r)$ (open ball).

Hint

$\overline{A}$ is the smallest closed set containing $A$ ; boundary $=\overline A\setminus\mathring A$ .

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Solution

Definitions.
- Interior: union of open subsets of $A$ .
- Closure: intersection of closed sets containing $A$ ; equivalently $A$ plus its limit points.
- Boundary: $\overline A\setminus\mathring A$ .

For $A=B(a,r)$ .
- Interior: $A$ itself (it is open).
- Closure: $\overline B(a,r)=\{x:\|x-a\|\le r\}$ .
- Boundary: sphere $\{x:\|x-a\|=r\}$ .

Conclusion. For an open ball, boundary is the sphere, closure is the closed ball.

Interior, closure, boundary

Statement

Solution

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