# Examples of Kuratowski closure axioms translation into the language of interior operators?

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?

• Depending on the situation you're working in, defining a closure operator may be easier than defining an interior operator. One example might be in the world of proximity spaces (en.wikipedia.org/wiki/Proximity_space). The topology on a proximity space is usually defined by a closure operator. An interior operator can also be defined, but it isn't quite as nice when doing the things one normally does with proximity spaces (deal with compactifications, etc.). Commented May 30, 2019 at 15:09

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:
1. $$f(X)=X$$.
2. $$\forall A \subseteq X: f(f(A))=f(A)$$.
3. $$\forall A,B \subseteq X: f(A \cap B)=f(A) \cap f(B)$$.
4. $$\forall A \subseteq X: f(A) \subseteq A$$.
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.