An exercise on topologies from interior operators

In a preceding entry of this blog, we saw that topologies arise from closure operators. This note raises the question of whether this is also valid for interior operators. The question is framed precisely in the exercise below. To approach the issue with serenity, it should be remembered here that closure and interior are dual concepts.

Exercise.

Let X be a set. If i:\mathcal{P}(X)\rightarrow\mathcal{P}(X) is an operator which carries subsets of X into subsets of X, and \mathfrak{O} the family of all subsets such that i(A)=A, under what conditions will \mathfrak{O} define a topology on X and i be the interior operator relative to this topology?

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