Let $S$ be a set. Then an interior operator is a function $\,^\circ \colon \mathcal{P}(S) \to \mathcal{P}(S)$ which satisfies the following properties:

If $S$ is a topological space, then the operator which assigns to each set its interior satisfies these axioms. Conversely, given an interior operator $\,^\circ$ on a set $S$ the set $\{X^\circ \mid X \subset S\}$ defines a topology on $S$ in which $X^\circ$ is the interior of $X$ for any subset $X$ of $S$ Thus, specifying an interior operator on a set is equivalent to specifying a topology on that set.

The concepts of interior operator and closure operator are closely related. Given an interior operator $\,^\circ$ one can define a closure operator $\,^c$ by the condition $$ X^c = ({(X')^\circ})\vphantom{X}' $$ and, given a closure operator $\,^c$ one can define an interior operator $\,^\circ$ by the condition $$ X^\circ = ({(X')^c})\vphantom{X}' .$$