Exercise 584Topology › Topology of ℝL1Added 3 Nov 2026

Arbitrary unions of open sets

#unions#open sets#arbitrary

Statement

Show that an arbitrary union of open sets of $\mathbb{R}$ is open.

Hint

For $x$ in the union, it belongs to at least one open set $U_i$ ; use the open ball around $x$ inside $U_i$ .

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Solution

Proof. Let $(U_i)_{i\in I}$ be open and $x\in\bigcup_i U_i$ . Pick $i_0$ with $x\in U_{i_0}$ . Since $U_{i_0}$ is open, there is $r>0$ with $B(x,r)\subseteq U_{i_0}\subseteq\bigcup_i U_i$ .

Conclusion. $\bigcup_i U_i$ is open.

Remark. Finite intersections of open sets are open, but infinite intersections may fail: $\bigcap_{n\ge1}]-\frac1n,\frac1n[=\{0\}$ , not open.

Arbitrary unions of open sets

Statement

Solution

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