Definition:Closure


 * Closure Operator
 * Definition:Closure of Set under Closure Operator
 * Definition:Closure of Element under Closure Operator


 * Order Theory:
 * Upper Closure
 * Lower Closure


 * Abstract Algebra:
 * Closure: An algebraic structure $\struct {S, \circ}$ has the property of closure $\forall \tuple {x, y} \in S \times S: x \circ y \in S$.
 * Integral Closure: The set of all elements of $A$ (where $A / R$ is a ring extension) that are integral over $R$.
 * Definition:Normal Closure of Field Extension
 * Definition:Normal Closure of Subset of Group


 * Topology:
 * Closure: The closure of a subset $A$ of a topological space $T$ is the union of $A$ and its boundary.
 * Closure: The closure of a subset $H$ of a metric space $M$ is the union of the isolated points of $H$ and all limit points of $H$.


 * Normed Vector Space:
 * Closure


 * Set Theory:
 * The transitive closure of a set $S$ is the smallest transitive superset of $S$.


 * Relation Theory:
 * The reflexive closure $\RR^=$ of a relation $\RR$ on $S$ is the smallest reflexive relation on $S$ which contains $\RR$.
 * The symmetric closure $\RR^\leftrightarrow$ of a relation $\RR$ on $S$ is the smallest symmetric relation on $S$ which contains $\RR$.
 * The transitive closure $\RR^+$ of a relation $\RR$ on $S$ is the smallest transitive relation on $S$ which contains $\RR$.

Also see

 * Definition:Closed
 * Definition:Saturation