Closure (mathematics)

In mathematics, especially algebra, is meant by closure of an amount with respect to a link that selecting any combination of elements of this set again yields an element of the set. For example, the set of integers is closed for addition, subtraction and multiplication, but not with respect to the division. Algebraic structures with multiple links is considered according to the seclusion respect of all these links.

Definition

Be a digit inner join on a set, ie is a function. A non-empty subset is now called closed with respect to when

Applies to all. This means restricted to the domain of a digit inner join on must be again.

Examples

• A subgroup is a non-empty subset of a group which is closed with respect to the link and the inverses.
• A subspace is a non-empty subset of a vector space that is closed under vector addition and scalar multiplication.
• In general, a sub-structure algebraic a ( non-empty ) subset of an algebraic structure is completed with respect to all the links of this structure.

The importance of isolation with respect to a link can be best understood if we consider examples in which it is violated.

• So, as a structure of the group is not completed, so no subgroup. This subset is indeed closed for addition, but not with respect to the Inverse Education: with does not belong to.
• The intersection of two subspaces of a vector space is always itself a subspace, but the union of two subspaces is not necessarily a subspace. The association is indeed closed under the scalar multiplication, but not necessarily with respect to the vector addition.

Generalization

Analogously, a subset also completed over one - digit inner join when their image is.

Example:

• Is on the power set of an infinite set and the set of all closed sets with respect to a T1 - topology, that is, contains all the ( infinitely many ) one-element subsets of, then is a closed set with respect to the set-theoretic average.

The property that a link on a quantity always delivers uniquely determined values ​​, also referred to as well- definedness of this link.