Let $X$ be a topological space and $A$ be a disconnected subspace of $X$. Is there an example such that there does not exist a pair of nonempty open subsets $U,V$ in $X$ satisfying $U\cap V=\emptyset$ and $(U\cup V)\cap A=A$?
I was marking general topology homework and I found one student assume the existence of such a pair $(U,V)$, but I think this is false in general. (It should be assumed that $U\cap V\cap A=\emptyset$ instead of $U\cap V=\emptyset$ to gurantee the existence of such $(U,V)$). However, I am having trouble finding a counterexample for this.