Definition:Closed


 * Topology:
 * Closed set: In topology, a subset of a topological space which contains all its limit points.
 * Closed mapping: In topology, a mapping which maps closed sets to closed sets.


 * Analysis:
 * Closed interval: An interval which includes its endpoints.


 * Graph Theory:
 * Closed walk: A walk whose first vertex is the same as the last.


 * Abstract Algebra
 * An algebraic structure $\left({S, \circ}\right)$ is closed iff $\forall \left({x, y}\right) \in S \times S: x \circ y \in S$.
 * A subset $T \subseteq S$ of an $R$-algebraic structure $\left({S, \circ}\right)_R$ is closed for scalar product iff $\forall \lambda \in R: \forall x \in T: \lambda \circ x \in T$.


 * Commutative Algebra
 * A commutative ring with unity $R$ is integrally closed in $A$ (where $A/R$ is a extension) if it equals its integral closure.
 * A subset $S$ of a commutative ring with unity is multiplicatively closed if $1 \in S$ and $\forall x, y \in S: x y \in S$.


 * Also see closure.