Definition:Closed

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:

• 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.
  • 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.

• 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 $\forall \lambda \in R: \forall x \in T: \lambda \circ x \in T$.
  • 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 commutative ring with unity is multiplicatively closed if $1 \in S$ and $\forall x, y \in S: x y \in S$.

• Also see closure.