# Definition:Closure Axioms (Matroid)

## Definition

Let $S$ be a finite set.

Let $\sigma : \powerset S \to \powerset S$ be a mapping where $\powerset S$ denotes the power set of $S$.

$\sigma$ is said to satisfy the closure axioms if and only if:

 $(\text S 1)$ $:$ $\ds \forall X \in \powerset S:$ $\ds X \subseteq \map \sigma X$ $(\text S 2)$ $:$ $\ds \forall X, Y \in \powerset S:$ $\ds X \subseteq Y \implies \map \sigma X \subseteq \map\sigma Y$ $(\text S 3)$ $:$ $\ds \forall X \in \powerset S:$ $\ds \map \sigma X = \map \sigma {\map \sigma X}$ $(\text S 4)$ $:$ $\ds \forall X \in \powerset S \land x, y \in S:$ $\ds y \not \in \map \sigma X \land y \in \map \sigma {X \cup \set x} \implies x \in \map \sigma {X \cup \set y}$

 It has been suggested that this page or section be merged into Definition:Closure Operator. (Discuss)