I am reading through Stephen Woolard's General topology and I am not sure what he means by "compatible".
The context is as follows. $X$ is a topological space space, and $\delta$ is a proximity on $X$. I am confused by what is meant when he says:
$\delta$ is compatible with the topology on $X$.
Does he mean that for any two subsets $A,B\subseteq X$ we have $A\delta B \iff \overline{A}\cap \overline{B} \neq \emptyset$? (here $\overline{A}$ is the closure of $A$ in the topology).
He means the following: a proximity $\delta$ induces a closure operator: $\overline{A} = \{x: \{x\}\delta A\}$ and this defines a topology in the usual way (closed sets are sets equal to their closure, open sets are their complement etc. ) If this topology equals the original topology on $X$, this topology is compatible with the proximity.