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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.
By comparison, in a general topological space, given sets A,B it is meaningful to say that a point x is arbitrarily close to A (i.e., in the closure of A), or perhaps that A is a smaller neighborhood of x than B, but notions of closeness of points and relative closeness are not described well by topological structure alone.
Covers are commonly used in the context of topology.If the set is a topological space, then a cover of is a collection of subsets {} of whose union is the whole space .In this case is said to cover , or that the sets cover .
The first formal definition of covering dimension was given by Eduard Čech, based on an earlier result of Henri Lebesgue. [4] A modern definition is as follows. An open cover of a topological space X is a family of open sets U α such that their union is the whole space, U α = X.
Functional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure (e.g. inner product, norm, topology, etc.) and the linear operators acting upon these spaces and respecting these structures in a suitable sense.
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 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
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.