In topology

Topology

Topology is a major area of mathematics concerned with properties that are preserved under continuous deformations of objects, such as deformations that involve stretching, but no tearing or gluing...

 and related areas in mathematics

Mathematics

Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

 closeness is one of the basic concepts in a topological space

Topological space

Topological spaces are mathematical structures that allow the formal definition of concepts such as convergence, connectedness, and continuity. They appear in virtually every branch of modern mathematics and are a central unifying notion...

. 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

Metric space

In mathematics, a metric space is a set where a notion of distance between elements of the set is defined.The metric space which most closely corresponds to our intuitive understanding of space is the 3-dimensional Euclidean 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

Closure operator

In mathematics, a closure operator on a set S is a function cl: P → P from the power set of S to itself which satisfies the following conditions for all sets X,Y ⊆ S....

 closes a given set by mapping it to a closed set

Closed set

In geometry, topology, and related branches of mathematics, a closed set is a set whose complement is an open set. In a topological space, a closed set can be defined as a set which contains all its limit points...

 which contains the original set and all points close to it. The concept of closeness is related to limit point

Limit point

In mathematics, a limit point of a set S in a topological space X is a point x in X that can be "approximated" by points of S in the sense that every neighbourhood of x with respect to the topology on X also contains a point of S other than x itself. Note that x does not have to be an element of S...

.

Definition

Given a metric space

Metric space

In mathematics, a metric space is a set where a notion of distance between elements of the set is defined.The metric space which most closely corresponds to our intuitive understanding of space is the 3-dimensional Euclidean 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

• if a point is close to a set and a set then and are close (the converse
  Converse (logic)
  In logic, the converse of a categorical or implicational statement is the result of reversing its two parts. For the implication P → Q, the converse is Q → P. For the categorical proposition All S is P, the converse is All P is S. In neither case does the converse necessarily follow from...
  
   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

Let and be two sets and a point.

• 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

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

Uniform space

In the mathematical field of topology, a uniform space is a set with a uniform structure. Uniform spaces are topological spaces with additional structure which is used to define uniform properties such as completeness, uniform continuity and uniform convergence.The conceptual difference between...

, 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.