Indicator function defined on the empty set

Question

Let $G \subset X$ be sets and $I_{G}$ be an indicator function, i.e. $I_{G}(x) := 1$ if $x \in G$ and $I_{G}(x) := 0$ if $x \notin G.$

But does the above definition cover the case where $G$ is empty?

Answer 1

Clearly, the function will just always be $0$ for any value $x$ in the domain. (Since $x\notin \emptyset \quad \forall x\in D$ for any domain $D$)

This is the definition of the empty set if you want. So yes, to answer your question the indicator is well defined when $G=\emptyset$.

$$I_\emptyset:D \to \{0,1\}$$ where $I_\emptyset (x) = 0 \quad \forall x \in D$ defines the function exactly for any set $D$.