Proximity space

This is a variation of topological space. View other variations of topological space

Definition

A proximity space is a set $X$ along with a binary relation $\delta$ on the power set of $X$ (called a proximity relation or nearness relation) satisfying the following conditions (note that we say that $A$ and $B$ are near, or $A\delta B$, if they are related, and we say that $A$ and $B$ are separated, or $A\not \delta B$ if $A$ and $B$ are not related):

1. Intersecting subsets are near: If $A\cap B\neq \varnothing$, then $A\delta B$. In other words, any two intersecting subsets are near.
2. Near implies nonempty: The empty set is not near to any set. In other words, $A\delta B$ implies that both $A$ and $B$ are nonempty.
3. Symmetry: $A\delta B\iff B\delta A$.
4. Distributivity: $A\delta (B\cup C)$ if and only if $A\delta B$ or $A\delta C$.
5. Separation: If $A\not \delta B$, there exists a set $C$ such that $A\not \delta C$ and $B\not \delta (X\setminus C)$.