From "An invitation to General Algebra and Universal Constructions" of George M. Bergman.
(i) Let $X$ be a set, $S = T = P(X)$, the set of all subsets of $X$, and let $R$ be the relation of having nonempty intersection. Since this is a symmetric relation, the two closure operators it induces are the same. Show that this operator $∗∗$ takes $A \subseteq P(X)$ to the set of those subsets of $X$ that contain a member of $A$. Deduce that under this Galois connection, the closed sets which are completely join-irreducible (cannot be written as a finite or infinite join of strictly smaller closed sets) are in natural one-to-one correspondence with the elements of $P(X)$, and that the general closed sets are precisely the unions of such closed sets.
(ii) Suppose $X$ is a topological space, $S = T = \{ open \ subsets \ of \ X \}$, and again let $R$ be the relation of having nonempty intersection. Can you characterize the resulting closure operator in this case? Can you get analogs of the remaining statements of part (i)?
I can't understand what (i) is meaning with "the operator $∗∗$ takes $A \subseteq P(X)$ to the set of those subsets of that contain a member of ". For example taking $X=\{1, 2, 3, 4\}$ and $A=\{\{1,2\}, \{2,3\}\}$. Every element of $A^*$ must contain an element of the the intersection of all the elements of $A$, so must contain $2$. $A^*=\{\{2\}, \{1, 2\}, \{2, 3\}, \{2, 4\}, \{1, 2, 3\}, \{2, 3, 4\}, \{1, 2, 3, 4\} \}$ and $A^{**}=A^*$.
What am I doing wrong?