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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 .
Closure operators allow generalizing the concept of closure to any partially ordered set. Given a poset S whose partial order is denoted with ≤ , a closure operator on S is a function C : S → S {\displaystyle C:S\to S} that is
If [K1] is omitted instead, then an operator satisfying [K2], [K3] and [K4'] is said to be a Moore closure operator. [7] A pair ( X , c ) {\displaystyle (X,\mathbf {c} )} is called Kuratowski , Čech or Moore closure space depending on the axioms satisfied by c {\displaystyle \mathbf {c} } .
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 ...
It includes closure operators that can also be found in the parent category, or in diffusing subcategories of the parent. This is a non-diffusing subcategory of Category:General topology . It includes concepts that can also be found in the parent category, or in diffusing subcategories of the parent.
Thus the theory of interior algebras may be formulated using the closure operator instead of the interior operator, in which case one considers closure algebras of the form S, ·, +, ′, 0, 1, C , where S, ·, +, ′, 0, 1 is again a Boolean algebra and C satisfies the above identities for the closure operator. Closure and interior algebras ...
Given a linear operator A, not necessarily closed, if the closure of its graph in X ⊕ Y happens to be the graph of some operator, that operator is called the closure of A, and we say that A is closable. Denote the closure of A by A. It follows that A is the restriction of A to D(A).
Closure operator definition [ edit ] Instead of specifying the open subsets of a topological space, the topology can also be determined by a closure operator (denoted cl), which assigns to any subset A ⊆ X its closure , or an interior operator (denoted int), which assigns to any subset A of X its interior .