| context | $X$ |
| context | $A\subseteq X$ |
| definiendum | $\chi_A:X\to \{0,1\} $ |
| definiendum | $ \chi_A(x) := \begin{cases} 1 & \mathrm{if}\ x\in A\\\\ 0 & \mathrm{else} \end{cases}$ |
The complement $A^c$ is taken w.r.t. $X$.
The symbols “$0,1$” here are essentially placeholders. One could choose any set of two distinct elements as codomain.
Wikipedia: Characteristic function