Definition:Closed


 * Topology: A set can be closed in a number of contexts:
 * Closed set: In topology, a subset of a topological space which contains all its limit points.
 * Closed set: In a metric space: a set which contains all its limit points.


 * Other uses of closed in Topology:
 * Regular closed set: a set which equals the closure of its interior.
 * Closed mapping: In topology, a mapping which maps closed sets to closed sets.
 * Closed extension topology: The set of all sets formed by adding a point $p$ to all the open sets of a given topology and then including the empty set.


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


 * Analysis:
 * Closed real interval: A real 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 $\forall \lambda \in R: \forall x \in T: \lambda \circ x \in T$.
 * A field $K$ is algebraically closed if the only algebraic extension of $K$ is $K$ itself.


 * 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 ring with unity is multiplicatively closed if $1 \in S$ and $\forall x, y \in S: x y \in S$.

Also see

 * Definition:Closure


 * Note that a closed subset can be taken to mean either of:
 * a closed set in topology
 * a subset $T$ of an algebraic structure $\left({S, \circ}\right)$ which is closed under $\circ$; such a subset is also known as a submagma.