Open disc

The open disc with centre $z$ and radius $r>0$ is $\{w:|w-z|<r\}$.

The unit disc in the complex plane is $\mathbb{D}=\{z\in \mathbb{C}:|z|<1\}$.

A biholomorphism of the unit disk to itself. Every such map has the form $e^{i\theta }(z-a)/(1-\overline{a}z)$ with $|a|<1$. These maps normalize an extremal univalent function in the proof of the Riemann mapping theorem.

A disc consists of the points whose distance from a centre is below a fixed radius, with the boundary included or excluded according to context.

A closed $n$-disc is a space homeomorphic to $D^n=\{x\in {\mathbb{R}}^n:|x|\le 1\}$; its boundary is the sphere $S^{n-1}$.