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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.
In mathematics, a subset of a given set is closed under an operation of the larger set if performing that operation on members of the subset always produces a member of that subset. For example, the natural numbers are closed under addition, but not under subtraction: 1 − 2 is not a natural number, although both 1 and 2 are.
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 ...
In the classic definition of the closeness centrality, the spread of information is modeled by the use of shortest paths. This model might not be the most realistic for all types of communication scenarios. Thus, related definitions have been discussed to measure closeness, like the random walk closeness centrality introduced by Noh and Rieger ...
The notion of closed set is defined above in terms of open sets, a concept that makes sense for topological spaces, as well as for other spaces that carry topological structures, such as metric spaces, differentiable manifolds, uniform spaces, and gauge spaces. Whether a set is closed depends on the space in which it is embedded.
In mathematics, a structure on a set (or on some sets) refers to providing it (or them) with certain additional features (e.g. an operation, relation, metric, or topology). Τhe additional features are attached or related to the set (or to the sets), so as to provide it (or them) with some additional meaning or significance.
By definition, the map : is a relatively closed map if and only if the surjection: is a strongly closed map. If in the open set definition of "continuous map" (which is the statement: "every preimage of an open set is open"), both instances of the word "open" are replaced with "closed" then the statement of results ("every preimage of a ...
From a spatial point of view, nearness (a.k.a. proximity) is considered a generalization of set intersection.For disjoint sets, a form of nearness set intersection is defined in terms of a set of objects (extracted from disjoint sets) that have similar features within some tolerance (see, e.g., §3 in).