# 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. 1.

if $A\cap B\neq\varnothing$, then $A\delta B$;

2. 2.

if $A\delta B$, then $A\neq\varnothing$ and $B\neq\varnothing$;

3. 3.

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

4. 4.

$(A_{1}\cup A_{2})\delta B$ iff $A_{1}\delta B$ or $A_{2}\delta B$;

5. 5.

$A\delta^{\prime}B$ implies the existence of $C\subseteq X$ with $A\delta^{\prime}C$ and $(X-C)\delta^{\prime}B$, 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 $A\cap B=\varnothing$. 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 $\varnothing^{c}=\varnothing$. 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 $x\delta^{\prime}C$ and $(X-C)\delta^{\prime}A$. The second part says that if $y\in X-C$, 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 non-empty set. For $A,B\subseteq X$, define $A\delta B$ iff $A\cap B\neq\varnothing$. 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 $A\neq\varnothing$ and $B\neq\varnothing$. Then $\delta$ is also a proximity. It is called the indiscrete proximity on $X$.

## References

• 1 S. Willard, General Topology, Addison-Wesley, 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 2013-03-22 16:48:11 Last modified on 2013-03-22 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