I read here
The abstract theory of closure operators and the Kuratowski closure axioms can be easily translated into the language of interior operators, by replacing sets with their complements.
I wish to see some examples.
Ok, I understand that
${\bar {S}}=X\backslash (X\backslash S)^{\circ }$
where X is the topological space containing S, and the backslash refers to the set-theoretic difference.
but this 'conversion' does not tell me much. For example, why use the interior operator theory rather than Kuratowski closure axioms ?
What do you gain?
Best Answer
The axioms for an interior operator $f$ on the family of subsets of $X$, are the dual ones to the closure axioms, so we have $f: \mathscr{P}(X) \to \mathscr{P}(X)$ with:
and such an $f$ is called an interior operator. There is no real advantage to using this vs using a closure operator. Here also, the translation is simple: for such an $f$, setting $c(A)=X\setminus f(X\setminus A)$ means that $c$ is a closure operator. Having one (an interior operator) means having the other (a closure operator) too, and vice versa. In practice there is more theory on closure (like) spaces, this has grown historically, and so the closure approach is most often seen in books. This is mostly a matter of choice and tradition.