Indicator vector

In mathematics, the indicator vector, characteristic vector, or incidence vector of a subset T of a set S is the vector $$x_T := (x_s)_{s\in S}$$ such that $$x_s = 1$$ if $$s \in T$$ and $$x_s = 0$$ if $$s \notin T.$$

If S is countable and its elements are numbered so that $$S = \{s_1,s_2,\ldots,s_n\}$$, then $$x_T = (x_1,x_2,\ldots,x_n)$$ where $$x_i = 1$$ if $$s_i \in T$$ and $$x_i = 0$$ if $$s_i \notin T.$$

To put it more simply, the indicator vector of T is a vector with one element for each element in S, with that element being one if the corresponding element of S is in T, and zero if it is not.

An indicator vector is a special (countable) case of an indicator function.

Example
If S is the set of natural numbers $$\mathbb{N}$$, and T is some subset of the natural numbers, then the indicator vector is naturally a single point in the Cantor space: that is, an infinite sequence of 1's and 0's, indicating membership, or lack thereof, in T. Such vectors commonly occur in the study of arithmetical hierarchy.