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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 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 ...
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).
Closeness may refer to: closeness (mathematics) closeness (graph theory), the shortest path between one vertex and another vertex; the personal distance between two people in proxemics; Social connectedness; Closeness (album), a 1976 album by Charlie Haden; Closeness, a 2017 Russian film
If is a group and is the free group on countably many generators, then by a finite set of equations and inequations with coefficients in we mean a pair of subsets and of the free product of and . This formalizes the notion of a set of equations and inequations consisting of variables x i {\displaystyle x_{i}\ } and elements g j {\displaystyle g ...
It follows that, in a specific example, when closeness is proved, there is no need to check the axioms for proving that a substructure is a structure of the same type. Given a subset X of an algebraic structure S, the closure of X is the smallest substructure of S that is closed under all operations of S.
In mathematics, a metric space is a set where a notion of distance (called a metric) between elements of the set is defined. Much of analysis happens in some metric space; the most commonly used are the real line , the complex plane , Euclidean space , other vector spaces , and the integers .
This definition is equivalent to the topological one, as applied to graphs, but it is easier to deal with in the context of graph theory. Graph theory also offers a context-free measure of connectedness, called the clustering coefficient. Other fields of mathematics are concerned with objects that are rarely considered as topological spaces.