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 
.
