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, i.e, if $C, D$ in $B$ and $C\cap D=\emptyset$, then $clC\cap clD=\emptyset$. Now, does $R$(real nubmer) with usual topology is a relative extremely disconnected space?

Actually real open intervals with rational left endpoint and irrational right endpoint are a base with that property. 

