Normality and paranormality in product spaces

Let βω denote the Stone–Čech compactification of the countable discrete space ω. We show that if p is a point of βω⧹ω, then all subspaces of (ω∪{p})×ω1 are paranormal, where (ω∪{p}) is considered as a subspace of βω. This answers a van Douwenʹs question. Moreover we show that the existence of a paranormal non-normal subspace of (ω+1)×ω1 is independent of ZFC, where ω+1 is the ordinal space {0,1,2,…,ω} with the usual order topology.