Set closure under a binary operator 
means that applying the operator to two elements
of a set 
returns a value which is itself a member of 
.
In a topological space, the topological closure of a set 
is the smallest closed set containing
. Closed sets
are closed under arbitrary intersection, so it is
also the intersection of all closed
sets containing 
.
Typically, it is just 
with all of its accumulation points.
The term "closure" is also used to refer to a "closed" version of a given set. The closure of a set can be defined in several equivalent ways, including
1. The set together with its accumulation points. Equivalently, it is the set together with its boundary. The boundary, also called the frontier, is generally different from the set closure.
2. The unique smallest closed set containing the given set.
3. The complement of the interior of the complement of the set.
4. The collection of all points such that every neighborhood of these points intersects the original set in a nonempty set.
In topologies where the T2-separation axiom is assumed, the closure of a finite set 
is 
itself.
