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 $(X-C){\delta }^{\prime }B$, 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 $A\cap B=\mathrm{\varnothing }$. 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 $(X-C){\delta }^{\prime }A$. 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 $A\cap B\ne \mathrm{\varnothing }$. Then $\delta$ so defined is a proximity on $X$, and is called the discrete proximity on $X$.

• indiscrete proximity. Again, $X$ is a non-empty 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 $X$.