Has anyone ever considered something like the following separation axiom?


$(\*)$ For any pair of open sets $O$ and $N$, there exist disjoint open sets containing $O\setminus N$ and $N\setminus O$.


Any $T\_1$ space that is not Hausdorff will fail $(\*)$ but I would be interested to know of any Hausdorff spaces failing $(\*)$. Such a space could not be too nice, as any perfectly normal (e.g. metric) space $X$ satisfies $(\*)$. To see this, take continuous $f,g:X\rightarrow\mathbb{R}\_+$ with $f^{-1}\{0\}=X\setminus O$ and $g^{-1}\{0\}=X\setminus N$ and consider the disjoint open sets on which $f-g$ is positive and negative. Can perfect normality be weakened here? Is $(\*)$ perhaps equivalent to some classical topological conditions?



Separating Differences of Open Sets
Note that condition $(\*)$ is equivalent to "every open subspace is normal" and it is a standard exercise to show that spaces with this property are exactly the completely normal ($T\_5$) spaces.


