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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.
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).
In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance.More specifically, a topological space is a set whose elements are called points, along with an additional structure called a topology, which can be defined as a set of neighbourhoods for each point that satisfy some axioms ...
Closeness may refer to: closeness (mathematics) closeness (graph theory), the shortest path between one vertex and another vertex;
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 ...
Closeness centrality, the total geodesic distance from a given vertex to all other vertices, is the best known example. [7] Note that this classification is independent of the type of walk counted (i.e. walk, trail, path, geodesic). Borgatti and Everett propose that this typology provides insight into how best to compare centrality measures.
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 ...
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.