Definition:Closure


 * Closure Operator


 * Order Theory:
 * Upper Closure
 * Lower 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 $R$.


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


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


 * 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

 * Definition:Closed
 * Definition:Saturation