proximity space
Let $X$ be a set. A binary relation^{} $\delta $ on $P(X)$, the power set^{} of $X$, is called a nearness relation on $X$ if it satisfies the following conditions: for $A,B\in P(X)$,

1.
if $A\cap B\ne \mathrm{\varnothing}$, then $A\delta B$;

2.
if $A\delta B$, then $A\ne \mathrm{\varnothing}$ and $B\ne \mathrm{\varnothing}$;

3.
(symmetry) if $A\delta B$, then $B\delta A$;

4.
$({A}_{1}\cup {A}_{2})\delta B$ iff ${A}_{1}\delta B$ or ${A}_{2}\delta B$;

5.
$A{\delta}^{\prime}B$ implies the existence of $C\subseteq X$ with $A{\delta}^{\prime}C$ and $(XC){\delta}^{\prime}B$, where $A{\delta}^{\prime}B$ means $(A,B)\notin \delta $.
If $x,y\in X$ and $A\subseteq X$, we write $x\delta A$ to mean $\{x\}\delta A$, and $x\delta y$ to mean $\{x\}\delta \{y\}$.
When $A\delta B$, we say that $A$ is $\delta $near, or just near $B$. $\delta $ is also called a proximity relation, or proximity for short. Condition 1 is equivalent^{} to saying if $A{\delta}^{\prime}B$, then $A\cap B=\mathrm{\varnothing}$. Condition 4 says that if $A$ is near $B$, then any superset^{} of $A$ is near $B$. Conversely, if $A$ is not near $B$, then no subset of $A$ is near $B$. In particular, if $x\in A$ and $A{\delta}^{\prime}B$, then $x{\delta}^{\prime}B$.
Definition. A set $X$ with a proximity as defined above is called a proximity space.
For any subset $A$ of $X$, define ${A}^{c}=\{x\in X\mid x\delta A\}$. Then ${}^{c}$ is a closure operator^{} on $X$:
Proof.
Clearly ${\mathrm{\varnothing}}^{c}=\mathrm{\varnothing}$. Also $A\subseteq {A}^{c}$ for any $A\subseteq X$. To see ${A}^{cc}={A}^{c}$, assume $x\delta {A}^{c}$, we want to show that $x\delta A$. If not, then there is $C\subseteq X$ such that $x{\delta}^{\prime}C$ and $(XC){\delta}^{\prime}A$. The second part says that if $y\in XC$, then $y{\delta}^{\prime}A$, which is equivalent to ${A}^{c}\subseteq C$. But $x{\delta}^{\prime}C$, so $x{\delta}^{\prime}{A}^{c}$. Finally, $x\in {(A\cup B)}^{c}$ iff $x\delta (A\cup B)$ iff $x\delta A$ or $x\delta B$ iff $x\in {A}^{c}$ or $x\in {B}^{c}$.∎
This turns $X$ into a topological space^{}. Thus any proximity space is a topological space induced by the closure operator defined above.
A proximity space is said to be separated if for any $x,y\in X$, $x\delta y$ implies $x=y$.
Examples.

•
Let $(X,d)$ be a pseudometric space. For any $x\in X$ and $A\subseteq X$, define $d(x,A):={inf}_{y\in A}d(x,y)$. Next, for $B\subseteq X$, define $d(A,B):={inf}_{x\in A}d(x,B)$. Finally, define $A\delta B$ iff $d(A,B)=0$. Then $\delta $ is a proximity and $(X,d)$ is a proximity space as a result.

•
discrete proximity. Let $X$ be a nonempty set. For $A,B\subseteq X$, define $A\delta B$ iff $A\cap B\ne \mathrm{\varnothing}$. Then $\delta $ so defined is a proximity on $X$, and is called the discrete proximity on $X$.

•
indiscrete proximity. Again, $X$ is a nonempty set and $A,B\subseteq X$. Define $A\delta B$ iff $A\ne \mathrm{\varnothing}$ and $B\ne \mathrm{\varnothing}$. Then $\delta $ is also a proximity. It is called the indiscrete proximity on $X$.
References
 1 S. Willard, General Topology, AddisonWesley, Publishing Company, 1970.
 2 S.A. Naimpally, B.D. Warrack, Proximity Spaces, Cambridge University Press, 1970.
Title  proximity space 
Canonical name  ProximitySpace 
Date of creation  20130322 16:48:11 
Last modified on  20130322 16:48:11 
Owner  CWoo (3771) 
Last modified by  CWoo (3771) 
Numerical id  17 
Author  CWoo (3771) 
Entry type  Definition 
Classification  msc 54E05 
Synonym  near 
Synonym  proximity 
Synonym  proximity relation 
Defines  nearness relation 
Defines  separated proximity space 
Defines  discrete proximity 
Defines  indiscrete proximity 