Characteristic function

Let $S$ be a set. For every subset $A\subseteq S$, its characteristic function, $f_A:S\to \{0,1\}$ is defined for all $x\in S$ by: $$ f_A(x) = \begin{cases} 1, &\text{if } x \in A, \\ 0, &\text{if } x \notin A \end{cases} $$

If $A\subseteq \mathbb{N}$ then $f_a:\mathbb{N}\to \{0,1\}$ can be represented by an infinite 0-1 sequence.

For example, given $\text{Odd }\subseteq \mathbb{N}$, $f_{\text{odd}}=0101010101[..]$.