Exercise 583Topology › Topology of ℝL1Added 3 Nov 2026

Finite intersections of open sets

#intersections#open sets#finite

Statement

Show that the intersection of finitely many open sets of $\mathbb{R}$ is open.

Hint

For $x\in U_1\cap\cdots\cap U_n$ , take the minimum of the radii from each $U_i$ .

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Solution

Proof. Let $U_1,\ldots,U_n$ be open and $x\in\bigcap U_i$ . For each $i$ , there is $r_i>0$ with $B(x,r_i)\subseteq U_i$ . Let $r=\min(r_1,\ldots,r_n)>0$ . Then $B(x,r)\subseteq\bigcap U_i$ .

Conclusion. The intersection is open.

Finite intersections of open sets

Statement

Solution

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