Proximity induced by a topology Let $\operatorname{cl}$ is the closure operator for some topology.
I will call induced proximity the proximity defined by the formula:
$$A\delta B\Leftrightarrow \operatorname{cl}(A)\cap\operatorname{cl}(B)\ne\varnothing.$$
Is induced proximity really a proximity for every given topological space?
Also: What I call here induced proximity is the weakest proximity generating our topology, right?

From Wikipedia, I learn about proximities:

The resulting topology is always completely regular. 

Thus either induced proximity fails to be a proximity if the given topological space is not regular.
Or the generated topology may differ from the given topology.
The latter kind of failure occurs in the space $\{1,2\}$ with open sets $\emptyset$, $\{1\}$, $\{1,2\}$. Here, $\{2\}$ is closed in the given topology, but in the generated topology, the closure of $\{2\}$ is $\{x\mid \{x\}\delta\{2\}\}=\{1,2\}$  (because $\operatorname{cl}(\{1\})=\{1,2\}$).
