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 $X$.

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