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?