Closure operator only on a subset $\mathscr A$ of the power set of $X$

Question

I am studying Steven A.Gaal's "Point Set Topology",Academic press,1964,and on page 31 Theorem 2 says that ". . . an operator $\bar A$ is defined $\mathbf{on}$ a subset $\mathscr A$ of the power set of $X$ . . .". I am really frustrated by the word "on" and wondering if the operator is a map from $\mathscr A$ to the power set of $X$,or, it just maps the sets in $\mathscr A$ to $\mathscr A$ itself. Because subsequently,it says on (Kb.3) that "If $A$ and $\bar A$ belongs to $\mathscr A$,then $\bar{\bar A}=\bar A$." and obviously author Professor Gaal specified the condition "$\bar A\in \mathscr A$" to be satisfied,and he did not give a proof that this could be generally true(he may think it trivial...),this causes me to doubt the meaning of his mysterious "on". Thank you for helping me learn math,it would be impossible for a major of physics to pursue an interest in math without your helps and stackexchange.