Definition:Closure


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


 * Mapping Theory:
 * Closure under Mapping: A set $S$ is closed under a mapping $\phi$ iff every indexed set of $S$ that is in the domain of $\phi$, is mapped into $S$ by $\phi$.


 * Topology:
 * Closure: The closure of a subset $A$ of a topological space $X$ is the union of $A$ and its boundary.


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


 * Also see Closed.