Exercise 283Topology › Topology of ℝⁿL1Added 3 Nov 2026

Open balls in $\mathbb{R}^n$

#open ball#Euclidean norm#definition

Statement

Define the open ball $B(a,r)$ in $\mathbb{R}^n$ and show it is an open set.

Hint

For $x\in B(a,r)$ , take $\rho=r-\|x-a\|>0$ and use the triangle inequality.

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Solution

Definition. $B(a,r)=\{x\in\mathbb{R}^n:\|x-a\|<r\}$ .

Openness. Let $x\in B(a,r)$ and $\rho=r-\|x-a\|>0$ . For $y\in B(x,\rho)$ : $\|y-a\|\le\|y-x\|+\|x-a\|<\rho+\|x-a\|=r$ . So $B(x,\rho)\subseteq B(a,r)$ .

Conclusion. Open balls are open sets.

Open balls in $\mathbb{R}^n$

Statement

Solution

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