A note on consonance of Gδ subsets

A space X is said to be consonant if, on the set of closed subsets of X, the upper Kuratowski topology coincides with the co-compact topology. It is known that Čech-complete spaces are consonant and that consonance is neither preserved by Gδ, subsets nor stable under products. We show that all Gδ subspaces of a consonant space X are consonant if the Vietoris topology on compact subsets of X is hereditarily Baire; and that is always the case if all compact subspaces of X are separable and of countable character in X. Spaces which are Gδ subspaces of consonant paracompact p-spaces are also shown to be consonant. Concerning products, we show that the product of a consonant paracompact p-space and a C̆ech-complete space is consonant. We also answer some questions of Nogura and Shakhmatov related to product and topological sum operations in the class of regular consonant spaces.