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