Closeness (mathematics)

From Wikipedia, the free encyclopedia
Jump to: navigation, search

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.

Note the difference between closeness, which describes the relation between two sets, and closedness, which describes a single set.

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[edit]

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

.

Similarly a set is called close to a set if

where

.

Properties[edit]

  • 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[edit]

Let and be two sets and a point.[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 either is close to or is close to
  • if is close to and for every point , is close to , then is close to .

Closeness relation between two sets[edit]

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[edit]

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[edit]

References[edit]

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