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

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

Closeness relation between a point and a set

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.
  • 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

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

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 and are called close to each other if they intersect all entourages, that is, for any entourage, is non-empty.