Is the following statement true? Connected components are open

Counterexample

Is this true?

Connected components are open

No — it is false.

The counterexample

$\mathbb{Q}$ with the usual topology

Why it fails

Components are singletons, which are not open. They are open exactly when the space is locally connected.

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.