PlanetMath: interior axioms

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interior axioms

(Definition)

Let 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 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 , the set $\{X^\circ \mid X \subset S\}$ defines a topology on in which $X^\circ$ is the interior of for any subset of . 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

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

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

$\displaystyle X^\circ = ({(X')^c})\vphantom{X}' .$

"interior axioms" is owned by rspuzio. [ full author list (2) ]

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