The product of two ordinals is hereditarily dually discrete

Let μ and ν be two ordinals. If X is a subspace of μ×ν, then X is dually discrete. This gives a positive answer to a question of Alas, Junqueira and Wilson. By this conclusion and a known conclusion we show that a subspace Y of μ×ν has countable spread if and only if the space Y is hereditarily a Lindelöf D-space.