# Definition:Closed

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## Disambiguation

This page lists articles associated with the same title. If an internal link led you here, you may wish to change the link to point directly to the intended article.

**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
- 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 $\phi$.
- Closed Set under Closure Operator
- 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.
- Closed path: A path whose first vertex is the same as the last, also known as a cycle.
- Closed trail: A trail whose first vertex is the same as the last, also known as a circuit.

- 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 $\forall x, y \in S: x y \in S$.