# Definition:Closure

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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.

**Closure** may refer to:

- 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 $R$.
- Definition:Normal Closure of Field Extension
- Definition:Normal Closure of Subset of Group

- 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 $\mathcal R$.