Exercise 281Topology › Topology of ℝL1Added 3 Nov 2026

Open intervals are open sets

#open sets#open intervals#definition

Statement

Show that any open interval $]a,b[$ (with $a<b$ ) is an open set of $\mathbb{R}$ .

Hint

For any $x\in]a,b[$ , take $r=\min(x-a,b-x)>0$ ; then $]x-r,x+r[\subseteq]a,b[$ .

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Radius. Let $x\in]a,b[$ and $r=\min(x-a,b-x)>0$ . Then for any $y\in]x-r,x+r[$ : $y<x+r\le b$ and $y>x-r\ge a$ . So $y\in]a,b[$ .

Conclusion. Every $x\in]a,b[$ has a neighbourhood inside $]a,b[$ ; the interval is open.