Concept
Closure operator
In mathematics, a closure operator on a set S is a function from the power set of S to itself that satisfies the following conditions for all sets {| border="0" |- |
| (cl is extensive), |
|---|
| (cl is increasing), |
| - |
| (cl is idempotent). |
| } |
| Closure operators are determined by their closed sets, i.e., by the sets of the form cl(X), since the closure cl(X) of a set X is the smallest closed set containing X. Such families of "closed sets" are sometimes called closure systems or "Moore families". A set together with a closure operator on it is sometimes called a closure space. Closure operators are also called "hull operators", which prevents confusion with the "closure operators" studied in topology. |
| E. H. Moore studied closure operators in his 1910 Introduction to a form of general analysis, whereas the concept of the closure of a subset originated in the work of Frigyes Riesz in connection with topological spaces. Though not formalized at the time, the idea of closure originated in the late 19th century with notable contributions by Ernst Schröder, Richard Dedekind and Georg Cantor. |
| 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 relative interior is not a closure operator: although it is idempotent, it is not increasing and if is a cube in and is one of its faces, then , but and , so it is not increasing. |
| In topology, the closure operators are topological closure operators, which must satisfy |
| for all (Note that for this gives ). |
| In algebra and logic, many closure operators are finitary closure operators, i.e. they satisfy |
| In the theory of partially ordered sets, which are important in theoretical computer science, closure operators have a more general definition that replaces with . (See . |
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Duality (order theory)
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