Set Closure

A set  and a binary operator  are said to exhibit closure if applying the binary operator to two elements  returns a value which is itself a member of .

The 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 plus its limit points, also called "boundary" points, the union of which is also called the "frontier."

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.

REFERENCES:

Croft, H. T.; Falconer, K. J.; and Guy, R. K. Unsolved Problems in Geometry. New York: Springer-Verlag, p. 2, 1991.