Is the following statement true? cl(A ∩ B) = cl(A) ∩ cl(B)

No. A = ℚ, B = ℝ∖ℚ in ℝ cl(A ∩ B) = cl(∅) = ∅ but cl A ∩ cl B = ℝ.

Counterexample

Is this true?

cl(A ∩ B) = cl(A) ∩ cl(B)

No — it is false.

The counterexample

$A = \mathbb{Q}, B = \mathbb{R}$ ∖ $\mathbb{Q}$ in $\mathbb{R}$

Why it fails

cl(A ∩ B) = cl(∅) = ∅ but cl A ∩ cl $B = \mathbb{R}$ .

Keep practicing

Hypothesis dropped

A theorem applied without checking one of its preconditions.

Published 23 Aug 2026. Found a mistake? Tell us — corrections are the fastest thing we act on.