Closure operator

In <a href="/facts/Mathematics/pxTouaz4">mathematics</a>, a closure operator on a <a href="/facts/Set_(mathematics)/BfucfHMq">set</a> S is a <a href="/facts/Function_(mathematics)/CdReEKCh">function</a> 
 
 
 
 cl
 :
 
 
 P
 
 
 (
 S
 )
 →
 
 
 P
 
 
 (
 S
 )
 
 
 {\displaystyle \operatorname {cl} :{\mathcal {P}}(S)\rightarrow {\mathcal {P}}(S)}
 
 from the <a href="/facts/Power_set/FVuf8PDD">power set</a> of S to itself that satisfies the following conditions for all sets 
 
 
 
 X
 ,
 Y
 ⊆
 S
 
 
 {\displaystyle X,Y\subseteq S}

<table><tbody><tr><td> X ⊆ cl ⁡ ( X ) {\displaystyle X\subseteq \operatorname {cl} (X)} </td><td>     (cl is extensive),</td></tr><tr><td> X ⊆ Y ⇒ cl ⁡ ( X ) ⊆ cl ⁡ ( Y ) {\displaystyle X\subseteq Y\Rightarrow \operatorname {cl} (X)\subseteq \operatorname {cl} (Y)} </td><td>     (cl is increasing),</td></tr><tr><td> cl ⁡ ( cl ⁡ ( X ) ) = cl ⁡ ( X ) {\displaystyle \operatorname {cl} (\operatorname {cl} (X))=\operatorname {cl} (X)} </td><td>     (cl is idempotent).</td></tr></tbody></table>
Closure operators are determined by their closed sets, i.e., by the sets of the form cl(X), since the closure cl(X) of a set X is the smallest closed set containing X. Such families of "closed sets" are sometimes called closure systems or "Moore families". A set together with a closure operator on it is sometimes called a closure space. Closure operators are also called "hull operators", which prevents confusion with the "closure operators" studied in <a href="/facts/Point-set_topology/ezC7B0yP">topology</a>.

Closure operator open-in-new

Closure operator