Proximity 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 $$A \cap B \ne \varnothing$$, 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 $$B \not \delta (X \setminus C)$$.