The characteristic function of a subset$U$ of some set$X$ is a function from $X$ to the set $TV$ of truth values (which classically is $TV = \{\bot,\top\}$) that takes $a$ in $X$ to the truth value of the statement that $a \in U$. That is,

$\chi_U(a) \;\Leftrightarrow\; a \in U ,$

where $\chi_U$ (also often $1_U$) is the characteristic function of $U$.

Of a subobject

More generally, the characteristic morphism of a subobject$U$ of some objects $X$ in a category with a subobject classifier$\Omega$ is the morphism from $X$ to $\Omega$ that classifies $U$; we have that