Results related to the Michael line

For a Hausdorff space (X, τ) with a topology τ and Y ⊂ X, let (X, τ(X, Y)) be the space X with the topology τ(X, Y) defined by {G ∪ B ¦ G ϵ τ, B ⊂ Y}. We prove that in case (X, τ) is paracompact and perfectly normal with Y metrizable, (X, τ(X; Y)) × (Y, τ ¦Y) is normal iff Y is Fσ in (X, τ). An example is also given showing that without perfect normality of (X, τ) this result is not true even if (X, τ) is hereditarily paracompact.