In mathematics, a closure operator on a set S is a function cl : P ( S ) → P ( S ) {\displaystyle \operatorname {cl} :{\mathcal {P}}(S)\rightarrow {\mathcal {P}}(S)} from the power set of S to itself that satisfies the following conditions for all sets X , Y ⊆ S {\displaystyle X,Y\subseteq S}
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.
History 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.
Closed sets The closed sets with respect to a closure operator on S form a subset C of the power set P(S). Any intersection of sets in C is again in C. In other words, C is a complete meet-subsemilattice of P(S). Conversely, if C ⊆ P(S) is closed under arbitrary intersections, then the function that associates to every subset X of S the smallest set Y ∈ C such that X ⊆ Y is a closure operator. There is a simple and fast algorithm for generating all closed sets of a given closure operator. A closure operator on a set is topological if and only if the set of closed sets is closed under finite unions, i.e., C is a meet-complete sublattice of P(S). Even for non-topological closure operators, C can be seen as having the structure of a lattice. (The join of two sets X,Y ⊆ P(S) being cl(X ∪ {\displaystyle \cup } Y).) But then C is not a sublattice of the lattice P(S). Given a finitary closure operator on a set, the closures of finite sets are exactly the compact elements of the set C of closed sets. It follows that C is an algebraic poset. Since C is also a lattice, it is often referred to as an algebraic lattice in this context. Conversely, if C is an algebraic poset, then the closure operator is finitary.
Pseudo-closed sets Each closure operator on a finite set S is uniquely determined by its images of its pseudo-closed sets. These are recursively defined: A set is pseudo-closed if it is not closed and contains the closure of each of its pseudo-closed proper subsets. Formally: P ⊆ S is pseudo-closed if and only if
P ≠ cl(P) and if Q ⊂ P is pseudo-closed, then cl(Q) ⊆ P.
Examples
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 f ¯ {\displaystyle {\overline {f}}} of a function f : E → R ∪ { ± ∞ } {\displaystyle f\colon E\to \mathbb {R} \cup \{\pm \infty \}} , where E {\displaystyle E} is e.g. a normed space, defined implicitly epi ( f ¯ ) = epi ( f ) ¯ {\displaystyle \operatorname {epi} ({\overline {f}})={\overline {\operatorname {epi} (f)}}} , where epi ( f ) {\displaystyle \operatorname {epi} (f)} is the epigraph of a function f {\displaystyle f} . The relative interior ri {\displaystyle \operatorname {ri} } is not a closure operator: although it is idempotent, it is not increasing and if C 1 {\displaystyle C_{1}} is a cube in R 3 {\displaystyle \mathbb {R} ^{3}} and C 2 {\displaystyle C_{2}} is one of its faces, then C 2 ⊂ C 1 {\displaystyle C_{2}\subset C_{1}} , but ri ( C 1 ) ≠ ∅ ≠ ri ( C 2 ) {\displaystyle \operatorname {ri} (C_{1})\neq \emptyset \neq \operatorname {ri} (C_{2})} and ri ( C 1 ) ∩ ri ( C 2 ) = ∅ {\displaystyle \operatorname {ri} (C_{1})\cap \operatorname {ri} (C_{2})=\emptyset } , so it is not increasing. In topology, the closure operators are topological closure operators, which must satisfy
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