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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 $A,B\in P(X)$,
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 $(XC)\delta^{{\prime}}A$. The second part says that if $y\in XC$, 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 nonempty 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 nonempty 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, AddisonWesley, Publishing Company, 1970.
 2 S.A. Naimpally, B.D. Warrack, Proximity Spaces, Cambridge University Press, 1970.
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Nearness space by porton ✓
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