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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 ...
Definition: We say that the function (resp. set-valued function) f is closable in X × Y if there exists a subset D ⊆ X containing S and a function (resp. set-valued function) F : D → Y whose graph is equal to the closure of the set Gr f in X × Y. Such an F is called a closure of f in X × Y, is denoted by f, and necessarily extends f.
Closure (topology) – All points and limit points in a subset of a topological space; Limit of a sequence – Value to which tends an infinite sequence; Limit point of a set – Cluster point in a topological space; Subsequential limit – The limit of some subsequence
Closeness is a basic concept in topology and related areas in mathematics.Intuitively, we say two sets are close if they are arbitrarily near to each other. The concept can be defined naturally in a metric space where a notion of distance between elements of the space is defined, but it can be generalized to topological spaces where we have no concrete way to measure distances.
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 .
For set-valued functions [ edit ] Closed graph theorem for set-valued functions [ 6 ] — For a Hausdorff compact range space Y {\displaystyle Y} , a set-valued function F : X → 2 Y {\displaystyle F:X\to 2^{Y}} has a closed graph if and only if it is upper hemicontinuous and F ( x ) is a closed set for all x ∈ X {\displaystyle x\in X} .
The usual proof of the closed graph theorem employs the open mapping theorem.It simply uses a general recipe of obtaining the closed graph theorem from the open mapping theorem; see closed graph theorem § Relation to the open mapping theorem (this deduction is formal and does not use linearity; the linearity is needed to appeal to the open mapping theorem which relies on the linearity.)
The transitive closure of a set. [1] The algebraic closure of a field. [2] The integral closure of an integral domain in a field that contains it. The radical of an ideal in a commutative ring. In geometry, the convex hull of a set S of points is the smallest convex set of which S is a subset. [3]
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