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In mathematics, a subset of a given set is closed under an operation of the larger set if performing that operation on members of the subset always produces a member of that subset. For example, the natural numbers are closed under addition, but not under subtraction: 1 − 2 is not a natural number, although both 1 and 2 are.
A theory may be referred to as a deductively closed theory to emphasize it is defined as a deductively closed set. [1] Deductive closure is a special case of the more general mathematical concept of closure — in particular, the deductive closure of is exactly the closure of with respect to the operation of logical consequence
In geometry, topology, and related branches of mathematics, a closed set is a set whose complement is an open set. [1] [2] In a topological space, a closed set can be defined as a set which contains all its limit points. In a complete metric space, a closed set is a set which is closed under the limit operation.
In point-set topology, Kuratowski's closure-complement problem asks for the largest number of distinct sets obtainable by repeatedly applying the set operations of closure and complement to a given starting subset of a topological space. The answer is 14. This result was first published by Kazimierz Kuratowski in 1922. [1]
Convex hull (red) of a polygon (yellow). The usual set closure from topology is a closure operator. Other examples include the linear span of a subset of a vector space, the convex hull or affine hull of a subset of a vector space or the lower semicontinuous hull ¯ of a function : {}, where is e.g. a normed space, defined implicitly (¯) = ¯, where is the epigraph of a function .
The definition of a point of closure of a set is closely related to the definition of a limit point of a set.The difference between the two definitions is subtle but important – namely, in the definition of a limit point of a set , every neighbourhood of must contain a point of other than itself, i.e., each neighbourhood of obviously has but it also must have a point of that is not equal to ...
Examples of multiplicative sets include: the set-theoretic complement of a prime ideal in a commutative ring; the set {1, x, x 2, x 3, ...}, where x is an element of a ring; the set of units of a ring; the set of non-zero-divisors in a ring; 1 + I for an ideal I; the Jordan–Pólya numbers, the multiplicative closure of the factorials.
Every algebraically closed group is simple. No algebraically closed group is finitely generated. An algebraically closed group cannot be recursively presented. A finitely generated group has a solvable word problem if and only if it can be embedded in every algebraically closed group. The proofs of these results are in general very complex.