Closure operator and set of fixpoint
There is an intimate connection between closure operators and complete lattices.

Given a closure operator $C$ on a set $X$, we can construct a complete lattice $L(C)$ as follows:


The points of the lattice are the fixpoints of $C$.
The meet of the lattice is $A \wedge B = A \cap B$.
The join of the lattice is $A \vee B = C(A \cup B)$.


Conversely, given a complete lattice $L$ of subsets of $X$, we can construct a closure operator $C:=C(L)$ on $X$ using the formula
$$ C(A) = \bigcap \{ B \in L : B \supseteq A \}. $$

It turns out that these two operators are inverses:
$C(L(C)) = C$ and $L(C(L)) = L$.

In particular, this shows that $C$ is determined by the set $L(C)$ of fixpoints, since
$$
C(A) = \bigcap \{B \in L(C) : B \supseteq A\}.
$$

(This information is taken from the sample chapter of M-solid varieties of algebras by Koppitz and Denecke.)