Editing Matroid (section)

=== Closure operators <span class="anchor" id="closure_ops_anchor"></span> ===
Let <math>M</math> be a matroid on a finite set <math> E</math>, with rank function <math>r</math> as above.  The ''closure'' or ''span'' <math> \operatorname{cl}(A) </math> of a subset <math>A</math> of <math>E</math> is the set
:<math> \operatorname{cl}(A) = \Bigl\{\ x \in E \mid r(A) = r\bigl( A \cup \{x\} \bigr) \Bigr\}</math>.
This defines a [[closure operator]] <math> \operatorname{cl}: \mathcal{P}(E) \mapsto \mathcal{P}(E) </math> where <math>\mathcal{P}</math> denotes the [[power set]], with the following properties:

* (C1) For all subsets <math> X </math> of <math> E</math>, <math> X \subseteq \operatorname{cl}(X)</math>. 

* (C2) For all subsets <math>X</math> of <math> E</math>, <math>\operatorname{cl}(X)= \operatorname{cl}\left( \operatorname{cl}\left( X \right) \right)</math>. 

* (C3) For all subsets <math>X</math> and <math>Y</math> of <math>E</math> with <math> X\subseteq Y</math>, <math>\operatorname{cl}(X)\subseteq \operatorname{cl}(Y)</math>.

* (C4) For all elements <math>a</math> and <math>b</math> from <math>E</math> and all subsets <math>Y</math> of <math>E</math>, if <math> a \in \operatorname{cl}( Y \cup \{b\}) \smallsetminus \operatorname{cl}(Y)</math> then <math> b \in \operatorname{cl}( Y \cup \{a\}) \smallsetminus \operatorname{cl}(Y)</math>. 

The first three of these properties are the defining properties of a closure operator. The fourth is sometimes called the ''[[Saunders Mac Lane|Mac Lane]]–[[Ernst Steinitz|Steinitz]] exchange property''. These properties may be taken as another definition of matroid: every function <math>\operatorname{cl}: \mathcal{P}(E)\to \mathcal{P}(E)</math> that obeys these properties determines a matroid.<ref name=w7-9/>