# 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:

 $(S1)$ $:$ $\displaystyle \forall X \in \powerset S:$ $\displaystyle X \subseteq \map \sigma X$ $(S2)$ $:$ $\displaystyle \forall X, Y \in \powerset S:$ $\displaystyle X \subseteq Y \implies \map \sigma X \subseteq \map\sigma Y$ $(S3)$ $:$ $\displaystyle \forall X \in \powerset S:$ $\displaystyle \map \sigma X = \map \sigma {\map \sigma X}$ $(S4)$ $:$ $\displaystyle \forall X \in \powerset S \land x, y \in S:$ $\displaystyle y \not \in \map \sigma X \land y \in \map \sigma {X \cup \set x} \implies x \in \map \sigma {X \cup \set y}$