A note on the class of paracompact spaces whose product with every paracompact space is paracompact

We prove that if X is a paracompact space which has a neighborhood assignment x → H x such that for each y ∈ X the closure of the set { x ∈ X : y ∈ H x } is compact then the products T × X , for every paracompact space T, and X ω are paracompact. The first result answers a problem of H. Junnila.