Exercise 280Topology › Topology of ℝL1Added 3 Nov 2026

Open balls in $\mathbb{R}$

#open ball#interval#topology

Statement

For $a\in\mathbb{R}$ and $r>0$ , describe the open ball $B(a,r)$ and show it is an open set.

Hint

$B(a,r)=\{x:|x-a|<r\}=]a-r,a+r[$ .

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Identification. $B(a,r)$ is the open interval $]a-r,a+r[$ .

Openness. Every point of $]a-r,a+r[$ has a small ball around it contained in $]a-r,a+r[$ (previous exercise).

Conclusion. Open balls are open sets; they are the building blocks of the topology of $\mathbb{R}$ .