Modulus map

Definition

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

$z\mapsto |z|$

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:

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

Facts

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