Name for “the set described by characteristic function f”

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?

• The support, or the preimage of 1. But in this very special case you should probably draw attemtion to the assumption. – Ian Aug 9 '18 at 17:16
• You should include an explicit statement of your terminology. Something along the lines of "We write $X_a$ to denote the subset of $X$ given by $\{x\in X\mid a(x)=1\}$." or something like that. – Asaf Karagila Aug 9 '18 at 18:39

This is called the support of $f$. It comes up quite a it in analysis.