gn.general topology – Example of a compact $alpha$-normal space which is not normal

Could you help me to find an example of a compact space which is $alpha$-normal, but not normal? The definition of $alpha$-normal is any disjoint closed subsets $A, B$ can be separated by disjoint open subsets $U, V$ such that $Ucap A$ is dense in $A$ and $Vcap B$ is dense in $B$.

Thank you.