Finite unions of weak -refinable spaces and products of ordinals

In this note, we show that if X is the union of a finite collection { X i : i = 1 , … , k } of weak θ ¯ -refinable subspaces and e ( X ) = ω , then X is a Lindelöf space. We also show that the product of two ordinals is dually discrete. The last conclusion gives a positive answer to a question of Alas, Junqueira and Wilson.