Given a set $A$, there is a function called its "characteristic function", usually denoted $\chi_A$, $\mathbf{1}_A$, $I_A$, or $K_A$, defined as follows:
$$ \chi_A(x) = \begin{cases} 1 &\text{if } x \in A, \\ 0 &\text{if } x \notin A. \end{cases} $$
Conversely, given some function $a$ such that for all $x$, $a(x)\in\{0,1\}$, we can use it as a characteristic function to construct a set:
$$?_a=\{x\mid a(x)=1\}$$
Does the set induced by $a$ have a commonly used name or symbol? Or is there even just a more concise notation to construct it?
This is called the support of $f$. It comes up quite a it in analysis.