Is the following statement true? A compact T₁ space is Hausdorff

No. An infinite set with the cofinite topology Compact and T₁ but any two non-empty open sets intersect.

Counterexample

Is this true?

A compact $T_{1}$ space is Hausdorff

No — it is false.

The counterexample

An infinite set with the cofinite topology

Why it fails

Compact and $T_{1}$ but any two non-empty open sets intersect.

Keep practicing

Property not inherited

A property assumed to pass to subobjects, quotients, or through a chain. It does not.

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