# interior axioms

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

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

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

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

Title | interior axioms |
---|---|

Canonical name | InteriorAxioms |

Date of creation | 2013-03-22 16:30:37 |

Last modified on | 2013-03-22 16:30:37 |

Owner | rspuzio (6075) |

Last modified by | rspuzio (6075) |

Numerical id | 8 |

Author | rspuzio (6075) |

Entry type | Definition |

Classification | msc 54A05 |

Related topic | GaloisConnection |

Defines | interior operator |