Indicator vector

In mathematics, the indicator vector or characteristic vector or incidence vector of a subset "T" of a set "S" is the vector formula_1 such that formula_2 if formula_3 and formula_4 if formula_5
If "S" is countable and its elements are numbered so that formula_6, then formula_7 where formula_8 if formula_9 and formula_10 if formula_11
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 formula_12, 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.