Closure operator について

| (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 "Moore families", in honor of E. H. Moore who studied closure operators in 1911.〔Blyth p.11〕 Closure operators are also called "hull operators", which prevents confusion with the "closure operators" studied in topology. A set together with a closure operator on it is sometimes called a closure system.
Closure operators have many applications:
In topology, the closure operators are ''topological'' closure operators, which must satisfy
:

\operatorname(X_1 \cup\dots\cup X_n) = \operatorname(X_1)\cup\dots\cup \operatorname(X_n)

for all

n\in\N

(Note that for

n=0

this gives

\operatorname(\varnothing)=\varnothing

).
In algebra and logic, many closure operators are finitary closure operators, i.e. they satisfy
:

\operatorname(X) = \bigcup\left\ Y \text \right\}.

In universal logic, closure operators are also known as consequence operators.
In the theory of partially ordered sets, which are important in theoretical computer science, closure operators have an alternative definition.
== Closure operators in topology ==
(詳細はtopological closure of a subset ''X'' of a topological space consists of all points ''y'' of the space, such that every neighbourhood of ''y'' contains a point of ''X''. The function that associates to every subset ''X'' its closure is a topological closure operator. Conversely, every topological closure operator on a set gives rise to a topological space whose closed sets are exactly the closed sets with respect to the closure operator.
For topological closure operators the second closure axiom (being increasing) is redundant.

抄文引用元・出典: フリー百科事典『ウィキペディア（Wikipedia）』
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