Normality and paranormality in product spaces

Abstract

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})× ω 1 are paranormal, where ( ω∪{ p}) is considered as a subspace of βω. This answers a van Douwen's question. Moreover we show that the existence of a paranormal non-normal subspace of ( ω+1)× ω 1 is independent of ZFC, where ω+1 is the ordinal space {0,1,2,…, ω} with the usual order topology.