A topological space $X$ is called relative extremely disconnected if it has a base $B$ (for open subsets) such that disjoint elements in $B$ have disjoint closure. Does it exist an infinite Hausdorff space $X$ which is not relative extremely disconnected?
Hello dear Ali. I think the answer is yes. consider the closed unit interval $I=[0,1]$, and define the set $K$ as follows:$$K=I\times I (0,1)\times (0)$$ roughly speaking eliminate the interval $(0,1)$ from the bottom of the unit square. Now we are to define the base of each point of $K$.
It is obvious to see that this new space is Hausdorff. But it is not relatively extremely disconnected. To see this consider any neighborhoods of $(0,0)$ and $(1,0)$. it is intuitive to see that the closure of these neighborhoods intersect each other in some point at the edge $x=\frac{1}{2}$. 

