Modulus map

Definition

The modulus map, denoted $|\cdot |$, is a map $\mathbb{C}\to [0,\mathrm{\infty })$, denoted by:

$z\mapsto \left|z\right|$

It is defined as follows. If $z=x+iy$ is a complex number with $x,y$ being respectively the real part and imaginary part, then:

$\left|z\right|=\sqrt{x^2+y^2}$

Facts

The modulus map is an open map with respect to the usual topologies on $\mathbb{C}$ and $[0,\mathrm{\infty })$. Note that it is not an open map if thought of as a map $\mathbb{C}\to \mathbb{R}$.