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Closeness is a basic concept intopology and related areas inmathematics. Intuitively, we say two sets are close if they are arbitrarily near to each other. The concept can be defined naturally in ametric space where a notion of distance between elements of the space is defined, but it can be generalized totopological spaces where we have no concrete way to measure distances.

Theclosure operatorcloses a given set by mapping it to aclosed set which contains the original set and all points close to it. The concept of closeness is related tolimit point.

Given ametric space${\displaystyle (X,d)}$ a point${\displaystyle p}$ is calledclose ornear to a set${\displaystyle A}$ if

${\displaystyle d(p,A)=0}$,

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

${\displaystyle d(p,A):=\underset{a\in A}{inf}d(p,a)}$

where inf stands forinfimum. Similarly a set${\displaystyle B}$ is calledclose to a set${\displaystyle A}$ if

${\displaystyle d(B,A)=0}$

where

${\displaystyle d(B,A):=\underset{b\in B}{inf}d(b,A)}$.

• if a point${\displaystyle p}$ is close to a set${\displaystyle A}$ and a set${\displaystyle B}$ then${\displaystyle A}$ and${\displaystyle B}$ are close, but theconverse is not true.
• closeness between a point and a set is preserved bycontinuous functions.
• closeness between two sets is preserved byuniformly continuous functions.

Let${\displaystyle V}$ be some set. A relation between the points of${\displaystyle V}$ and the subsets of${\displaystyle V}$ is a closeness relation if it satisfies the following conditions:

Let${\displaystyle A}$ and${\displaystyle B}$ be two subsets of${\displaystyle V}$ and${\displaystyle p}$ a point in${\displaystyle V}$.^[1]

• If${\displaystyle p\in A}$ then${\displaystyle p}$ is close to${\displaystyle A}$.
• if${\displaystyle p}$ is close to${\displaystyle A}$ then${\displaystyle A\ne \mathrm{\varnothing }}$.
• if${\displaystyle p}$ is close to${\displaystyle A}$ and${\displaystyle B\supset A}$ then${\displaystyle p}$ is close to${\displaystyle B}$.
• if${\displaystyle p}$ is close to${\displaystyle A\cup B}$ then${\displaystyle p}$ is close to${\displaystyle A}$ or${\displaystyle p}$ is close to${\displaystyle B}$.
• if${\displaystyle p}$ is close to${\displaystyle A}$ and for every point${\displaystyle a\in A}$,${\displaystyle a}$ is close to${\displaystyle B}$, then${\displaystyle p}$ is close to${\displaystyle B}$.

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

Let${\displaystyle A}$,${\displaystyle B}$ and${\displaystyle C}$ be sets.

• if${\displaystyle A}$ and${\displaystyle B}$ are close then${\displaystyle A\ne \mathrm{\varnothing }}$ and${\displaystyle B\ne \mathrm{\varnothing }}$.
• if${\displaystyle A}$ and${\displaystyle B}$ are close then${\displaystyle B}$ and${\displaystyle A}$ are close.
• if${\displaystyle A}$ and${\displaystyle B}$ are close and${\displaystyle B\subset C}$ then${\displaystyle A}$ and${\displaystyle C}$ are close.
• if${\displaystyle A}$ and${\displaystyle B\cup C}$ are close then either${\displaystyle A}$ and${\displaystyle B}$ are close or${\displaystyle A}$ and${\displaystyle C}$ are close.
• if${\displaystyle A\cap B\ne \mathrm{\varnothing }}$ then${\displaystyle A}$ and${\displaystyle B}$ 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${\displaystyle p}$,${\displaystyle p}$ is calledclose to a set${\displaystyle A}$ if${\displaystyle p\in \mathrm{cl}(A)=\overline{A}}$.

To define a closeness relation between two sets the topological structure is too weak and we have to use auniform structure. Given auniform space, sets${\displaystyle A}$ and${\displaystyle B}$ are calledclose to each other if they intersect allentourages, that is, for any entourage${\displaystyle U}$,${\displaystyle (A\times B)\cap U}$ is non-empty.

• Topological space
• Uniform space

• General topology

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