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Closure


The term "closure" has various meanings in mathematics.

The topological closure of a subset A of a topological space X is the smallest closed subset of X containing A.

If R is a binary relation on some set A, then R has reflexive, symmetric and transitive closures, each of which is the smallest relation on A, with the indicated property, containing R. Consequently, given any relation R on any set A, there is always a smallest equivalence relation on A containing R.

For some arbitrary property P of relations, the relation R need not have a P-closure, i.e., there need not be a smallest relation on A with the property P, and containing R. For example, it often happens that a relation does not have an antisymmetric closure.

In algebra, the algebraic closure of a field F is a field F^_ which can be said to be obtained from F by adjoining all elements algebraic over F.


This entry contributed by Rasmus Hedegaard

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Cite this as:

Hedegaard, Rasmus. "Closure." From MathWorld--A Wolfram Web Resource, created by Eric W. Weisstein. https://mathworld.wolfram.com/Closure.html

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