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 .