Understanding definition of Cover

Question

• A cover of a set ${\displaystyle X}$ is a collection of sets whose union contains ${\displaystyle X}$ as a subset. Formally, if

$ C=\lbrace U_{\alpha }:\alpha \in A\rbrace$.

My question is: What is the $A$ mean, is $A\subseteq X$?

Answer 1

$A$ is an arbitrary index set, meaning that the cover could contain finitely many, countably infinitely many, or uncountably infinitely many elements. This is more general than saying $ C = \{ U_i: i = 1, \ldots, n\} $, for instance.