# interior axioms

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:

###### Axiom 1.

$S^{\circ}=S$

###### Axiom 2.

For all $X\subset S$, one has $X^{\circ}\subseteq S$.

###### Axiom 3.

For all $X\subset S$, one has $(X^{\circ})^{\circ}=X^{\circ}$.

###### Axiom 4.

For all $X,Y\subset S$, one has $(X\cap Y)^{\circ}=X^{\circ}\cap Y^{\circ}$.

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

 $X^{c}=({(X^{\prime})^{\circ}})\vphantom{X}^{\prime}$

and, given a closure operator $\,{}^{c}$, one can define an interior operator $\,{}^{\circ}$ by the condition

 $X^{\circ}=({(X^{\prime})^{c}})\vphantom{X}^{\prime}.$
Title interior axioms InteriorAxioms 2013-03-22 16:30:37 2013-03-22 16:30:37 rspuzio (6075) rspuzio (6075) 8 rspuzio (6075) Definition msc 54A05 GaloisConnection interior operator