Closure

The term "closure" has various meanings in mathematics.

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

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

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

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