Proximity space

Remarks

Let $(X, d)$ be a metric space and let $A \mathrel{\delta} B$ mean $\inf \set{d(a, b) : a \in A, b \in B} = 0$, with $\inf \emptyset = \infty$. Clauses 1 to 4 follow directly. For clause 5, suppose that $A$ is not near $B$, so the distance $r$ between them is positive. Let $E = \set{x : d(x, A) \ge r/2}$. Then $A$ is not near $E$, and each point of $X \setminus E$ is at distance more than $r/2$ from $B$, so $B$ is not near $X \setminus E$. The relation that makes all pairs of subsets near fails clause 2, since it makes $\emptyset$ near $\emptyset$. A proximity defines a topology: the closure of $A$ is the set of points $x$ with $\set{x} \mathrel{\delta} A$. Many authors also ask that $\set{x} \mathrel{\delta} \set{y}$ imply $x = y$; such a proximity is separated.