interior axioms

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

Axiom 1.

$S^{\circ }=S$

Axiom 2.

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

Axiom 3.

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

Axiom 4.

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

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

The concepts of interior operator and closure operator are closelyrelated.Given an interior operator ${}^{\circ }$, one candefine a closure operator ${}^c$ by the condition

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

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

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