Closeness (mathematics)

Properties

• 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 (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

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 ${\displaystyle V}$.

• 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 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 ${\displaystyle A}$,${\displaystyle B}$ 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

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 ${\displaystyle p}$, ${\displaystyle p}$ is called close 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 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.