The classic example (given in Kelley'sGeneral Topology, and attributed by him to Dieudonné and Morse, independently) uses ordinal spaces. Let be the set of all ordinal numbers less than the first uncountable ordinal and let , each with the order topology. Then and are both normal, but is not. (Let and . Then A and B are closed and disjoint in but do not have disjoint open neighbourhoods. The proof is not that easy. You would do best to look it up in Kelley's book.)