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 $A\cap B\ne \varnothing$ , then $A\delta B$ ;
2. if $A\delta B$ , then $A\ne \varnothing$ and $B\ne \varnothing$ ;
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. $A\delta'B$ implies the existence of $C \subseteq X$ with $A\delta'C$ and $(X-C)\delta'B$ , where $A\delta'B$ means $(A,B)\notin \delta$ .

If $x,y\in X$ and $A\subseteq X$ , we write $x\delta A$ to mean $\lbrace x\rbrace \delta A$ , and $x\delta y$ to mean $\lbrace x\rbrace \delta \lbrace y \rbrace$ .

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 $A\delta'B$ , then $A\cap B=\varnothing$ . 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 $A\delta' B$ , then $x\delta'B$ .

Definition. A set $X$ with a proximity as defined above is called a proximity space.

For any subset $A$ of $X$ , define $A^c=\lbrace x\in X\mid x\delta A\rbrace$ . Then $^c$ is a closure operator on $X$ :

Proof. Clearly $\varnothing^c=\varnothing$ . 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 $x\delta'C$ and $(X-C)\delta'A$ . The second part says that if $y\in X-C$ , then $y\delta'A$ , which is equivalent to $A^c \subseteq C$ . But $x\delta'C$ , so $x\delta'A^c$ . 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 $x\in B^c$ .

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 non-empty set. For $A,B\subseteq X$ , define $A\delta B$ iff $A\cap B\ne\varnothing$ . Then $\delta$ so defined is a proximity on $X$ , and is called the discrete proximity on $X$ .
• indiscrete proximity. Again, $X$ is a non-empty set and $A,B\subseteq X$ . Define $A\delta B$ iff $A\ne \varnothing$ and $B\ne \varnothing$ . Then $\delta$ is also a proximity. It is called the indiscrete proximity on $X$ .

Bibliography

1

S. Willard, General Topology, Addison-Wesley, Publishing Company, 1970.

2

S.A. Naimpally, B.D. Warrack, Proximity Spaces, Cambridge University Press, 1970.