# Definition:Closed

## Disambiguation

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**Closed** may refer to:

- Predicate Logic: a closed statement is a statement which every variable appears as a bound occurrence.

- 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.
- Closed Path
- Definition:Closed Region

- Mapping Theory:
- Closed under Mapping: A set $S$ is closed under a mapping $\phi$ if and only if every indexed set of $S$ that is in the domain of $\phi$ is mapped into $S$ by $\phi$.
- Definition:Closed Set under Closure Operator
- Definition:Closed Element under Closure Operator
- Closed in Galois Connection

- 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 $\struct {S, \circ}$ is closed if and only if $\forall \tuple {x, y} \in S \times S: x \circ y \in S$.
- A subset $T \subseteq S$ of an $R$-algebraic structure $\struct {S, \circ}_R$ is closed for scalar product if and only if $\forall \lambda \in R: \forall x \in T: \lambda \circ x \in T$.
- 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 and only if it equals its integral closure.
- A subset $S$ of a ring with unity is multiplicatively closed if and only if $1 \in S$ and $\forall x, y \in S: x y \in S$.