Is the following statement true? ℂ and the unit disc are biholomorphic (both are simply connected)

No. Liouville's theorem A biholomorphism 𝔻 → ℂ would invert to a bounded entire function. ℂ is the sole exception in the Riemann mapping theorem.

Counterexample

Is this true?

$\mathbb{C}$ and the unit disc are biholomorphic (both are simply connected)

No — it is false.

The counterexample

Liouville's theorem

Why it fails

A biholomorphism 𝔻 $\to \mathbb{C}$ would invert to a bounded entire function. $\mathbb{C}$ is the sole exception in the Riemann mapping theorem.

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.