Is the following statement true? An arbitrary intersection of open sets is open

No. ∩_{n≥1}(−1/n, 1/n) = {0} Only finite intersections preserve openness.

Counterexample

Is this true?

An arbitrary intersection of open sets is open

No — it is false.

The counterexample

∩ $_{n\ge1}(-1/n$ , 1/n) = {0}

Why it fails

Only finite intersections preserve openness.

Keep practicing

Standard counterexample

There is a canonical object that settles this. Recognising it is the whole question.

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