What is the purpose/meaning of parametrizing a set by an index set

Question

My lecture notes define $B=\{x_i|i\in I\}$ as a basis for a vector space, parametrized by the index set $I$, and uses this notation for the rest of the chapter. But what is the purpose of this notation when we can label the ${x_i}$ by anything we want? Does it not mean the same as simply denoting the vectors by $\{x_i\},\ i=1,...,n$?

Answer 1

If the vector spaces are finite dimensional, there is not much conceptual difference between using $i\in I$ and using $1\leq i\leq n$ to index your bases. However, $i\in I$ allows for infinite dimensions, without changing your notation.