proximal neighborhood
Let $X$ be a set and $P(X)$ its power set^{}. Let $\ll $ be a binary relation^{} on $P(X)$ satisfying the
following conditions, for any $A,B\subseteq X$:

1.
$X\ll X$,

2.
$A\ll B$ implies $A\subseteq B$,

3.
$A\ll B$ and $C\ll D$ imply $A\cap C\ll B\cap D$,

4.
$A\ll B$ implies ${B}^{\prime}\ll {A}^{\prime}$ (${}^{\prime}$ is the complement^{} operator)

5.
$A\subseteq B\ll C\subseteq D$, then $A\ll D$, and

6.
if $A\ll B$, then there is $C\subseteq X$, such that $A\ll C\ll B$.
By 1 and 4, it is easy to see that $\mathrm{\varnothing}\ll \mathrm{\varnothing}$. Also, 3 and 4 show that $A\cup C\ll B\cup D$ whenever $A\ll B$ and $C\ll D$. So $\ll $ is a topogenous order, which means $\ll $ is transitive^{} and antisymmetric. Under this order relation, we say that $B$ is a proximal neighborhood of $A$ if $A\ll B$.
The reason why we call $B$ a “proximal” neighborhood^{} is due to the following:
Theorem 1.
Let $X$ be a set. The following are true.

•
Let $\ll $ be defined as above. Define a new relation $\delta $ on $P(X)$: $A{\delta}^{\prime}{B}^{\prime}$ iff $A\ll B$. Then $\delta $ so defined is a proximity relation, turning $X$ into a proximity space.

•
Conversely, let $(X,\delta )$ is a proximity space. Define a new relation $\ll $ on $P(X)$: $A\ll B$ iff $A{\delta}^{\prime}{B}^{\prime}$. Then $\ll $ satisfies the six properties above.
Proof.
Suppose first that $X$ and $\ll $ are defined as above. We will verify the individual nearness relation axioms of $\delta $ by proving their contrapositives in each case, except the last axiom:

1.
if $A{\delta}^{\prime}B$, then $A\ll {B}^{\prime}$, or $A\subseteq {B}^{\prime}$, so $A\cap B=\mathrm{\varnothing}$;

2.
suppose either $A=\mathrm{\varnothing}$ or $B=\mathrm{\varnothing}$. In either case, $A\ll {B}^{\prime}$, which means $A{\delta}^{\prime}B$;

3.
if $A{\delta}^{\prime}B$, then $A\ll {B}^{\prime}$, so ${B}^{\prime \prime}\ll {A}^{\prime}$, or $B\ll {A}^{\prime}$, or $B{\delta}^{\prime}A$;

4.
if ${A}_{1}{\delta}^{\prime}B$ and ${A}_{2}{\delta}^{\prime}B$, then ${A}_{1}\ll B$ and ${A}_{2}\ll B$, so $({A}_{1}\cup {A}_{2})\ll B$, or $({A}_{1}\cup {A}_{2}){\delta}^{\prime}B$;

5.
if $A{\delta}^{\prime}B$, then $A\ll {B}^{\prime}$. So there is $D\subseteq X$ with $A\ll D$ and $D\ll {B}^{\prime}$. Let $C={D}^{\prime}$. Then $A\ll {C}^{\prime}$ and ${C}^{\prime}\ll {B}^{\prime}$, or $A{\delta}^{\prime}C$ and ${C}^{\prime}{\delta}^{\prime}B$.
Next, suppose $(X,\delta )$ is a proximity space. We now verify the six properties of $\ll $ above.

1.
since $X{\delta}^{\prime}\mathrm{\varnothing}$, $X\ll {\mathrm{\varnothing}}^{\prime}$, or $X\ll X$;

2.
suppose $A{\delta}^{\prime}{B}^{\prime}$, then if $x\in A$, we have $x{\delta}^{\prime}{B}^{\prime}$, implying $x\cap {B}^{\prime}=\mathrm{\varnothing}$, or $x\in B$;

3.
if $A\ll B$ and $C\ll D$, then $A{\delta}^{\prime}{B}^{\prime}$ and $C{\delta}^{\prime}{D}^{\prime}$, which means $A{\delta}^{\prime}({B}^{\prime}\cup {D}^{\prime})$ and $C{\delta}^{\prime}({B}^{\prime}\cup {D}^{\prime})$, which together imply $(A\cap C){\delta}^{\prime}({B}^{\prime}\cup {D}^{\prime})$, or $(A\cap C)\delta {(B\cap D)}^{\prime}$, or $A\cap C\ll B\cap D$;

4.
if $A\ll B$, then $A{\delta}^{\prime}{B}^{\prime}$, so ${B}^{\prime}{\delta}^{\prime}A$ (as $\delta $ is symmetric, so is its complement), which is the same as ${B}^{\prime}{\delta}^{\prime}{A}^{\prime \prime}$, or ${B}^{\prime}\ll {A}^{\prime}$;

5.
if $A\delta {D}^{\prime}$, then $B\delta {C}^{\prime}$ (since $A\subseteq B$ and ${D}^{\prime}\subseteq {C}^{\prime}$), so $B{\ll}^{\prime}C$, a contradiction^{};

6.
if $A\ll B$, then $A{\delta}^{\prime}{B}^{\prime}$, so there is $D\subseteq X$ with $A{\delta}^{\prime}D$ and ${D}^{\prime}{\delta}^{\prime}{B}^{\prime}$. Define $C={D}^{\prime}$, then $A\ll C$ and $C\ll B$, as desired.
This completes^{} the proof. ∎
Because of the above, we see that a proximity space can be equivalently defined using the proximal neighborhood concept. To emphasize its relationship with $\delta $, a proximal neighborhood is also called a $\delta $neighbhorhood.
Furthermore, we have
Theorem 2.
if $B$ is a proximal neighborhood of $A$ in a proximity space $\mathrm{(}X\mathrm{,}\delta \mathrm{)}$, then $B$ is a (topological) neighborhood of $A$ under the topology^{} $\tau \mathit{}\mathrm{(}\delta \mathrm{)}$ induced by the proximity relation $\delta $. In other words, if $A\mathrm{\ll}B$, then $A\mathrm{\subseteq}{B}^{\mathrm{\circ}}$ and ${A}^{c}\mathrm{\subseteq}B$, where ${}^{\mathrm{\circ}}$ and ${}^{c}$ denote the interior and closure operators^{}.
Proof.
Since $A{\delta}^{\prime}{B}^{\prime}$, then $x{\delta}^{\prime}{B}^{\prime}$ whenever $x\in A$, which is the contrapositive of the statement: $x\in {A}^{\prime}$ whenever $x\delta {B}^{\prime}$, which is equivalent^{} to ${B}^{\prime c}\subseteq {A}^{\prime}$, or $A\subseteq {B}^{\circ}$. Furthermore, if $x\notin B$, then $x\in {B}^{\prime}$. But $A{\delta}^{\prime}{B}^{\prime}$ b assumption^{}. This implies $x{\delta}^{\prime}A$, which means $x\notin {A}^{c}$. Therefore ${A}^{c}\subseteq B$. ∎
Remark. However, not every $\tau (\delta )$neighborhood is a $\delta $neighborhood.
Title  proximal neighborhood 

Canonical name  ProximalNeighborhood 
Date of creation  20130322 16:58:25 
Last modified on  20130322 16:58:25 
Owner  CWoo (3771) 
Last modified by  CWoo (3771) 
Numerical id  7 
Author  CWoo (3771) 
Entry type  Definition 
Classification  msc 54E05 
Synonym  proximity neighborhood 
Synonym  $\delta $neighborhood 