Exercise 255Topology › Metric and normed spacesL1Added 3 Nov 2026

Open balls in a metric space

#open ball#metric#definition

Statement

Define the open ball $B(a,r)$ in a metric space and show it is open.

Hint

For $x\in B(a,r)$ , use $r-\|x-a\|>0$ .

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Definition. $B(a,r)=\{x\in X:d(x,a)<r\}$ .

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

Conclusion. $B(x,\rho)\subseteq B(a,r)$ : open balls are open sets.