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 $\operatorname{cl}(\\{1\\})=\\{1,2\\}$).