Closeness (mathematics)

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.

The closure operator closes a given set by mapping it to a closed set which contains the original set and all points close to it. The concept of closeness is related to limit point.

Definition

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Given a metric space   a point   is called close or near to a set   if

 ,

where the distance between a point and a set is defined as

 

where inf stands for infimum. Similarly a set   is called close to a set   if

 

where

 .

Properties

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  • if a point   is close to a set   and a set   then   and   are close (the converse is not true!).
  • closeness between a point and a set is preserved by continuous functions
  • closeness between two sets is preserved by uniformly continuous functions

Closeness relation between a point and a set

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Let   be some set. A relation between the points of   and the subsets of   is a closeness relation if it satisfies the following conditions:

Let   and   be two subsets of   and   a point in  .[1]

  • If   then   is close to  .
  • if   is close to   then  
  • if   is close to   and   then   is close to  
  • if   is close to   then   is close to   or   is close to  
  • if   is close to   and for every point  ,   is close to  , then   is close to  .

Topological spaces have a closeness relationship built into them: defining a point   to be close to a subset   if and only if   is in the closure of   satisfies the above conditions. Likewise, given a set with a closeness relation, defining a point   to be in the closure of a subset   if and only if   is close to   satisfies the Kuratowski closure axioms. Thus, defining a closeness relation on a set is exactly equivalent to defining a topology on that set.

Closeness relation between two sets

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Let  ,  and   be sets.

  • if   and   are close then   and  
  • if   and   are close then   and   are close
  • if   and   are close and   then   and   are close
  • if   and   are close then either   and   are close or   and   are close
  • if   then   and   are close

Generalized definition

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The closeness relation between a set and a point can be generalized to any topological space. Given a topological space and a point  ,   is called close to a set   if  .

To define a closeness relation between two sets the topological structure is too weak and we have to use a uniform structure. Given a uniform space, sets A and B are called close to each other if they intersect all entourages, that is, for any entourage U, (A×B)∩U is non-empty.

See also

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References

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  1. ^ Arkhangel'skii, A. V.; Pontryagin, L.S. General Topology I: Basic Concepts and Constructions, Dimension Theory. Encyclopaedia of Mathematical Sciences (Book 17), Springer 1990, p. 9.