Exercise 589Topology › Topology of ℝⁿL1Added 3 Nov 2026

Stability of open and closed sets

#unions#intersections#stability

Statement

Show that arbitrary unions and finite intersections of open sets in $\mathbb{R}^n$ are open; dual properties for closed sets.

Hint

For finite intersections, take the minimum of radii. For closed sets, use De Morgan's laws.

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Union. If $x\in\bigcup U_i$ , pick $i_0$ with $x\in U_{i_0}$ ; use the ball around $x$ inside $U_{i_0}$ .

Finite intersection. With $x\in U_1\cap\cdots\cap U_n$ , take $r=\min_i r_i>0$ where $B(x,r_i)\subseteq U_i$ .

Closed sets. By De Morgan: arbitrary intersections and finite unions of closed sets are closed.

Conclusion. The topology is stable under the expected operations.