H-trivial spaces

Let X be a regular Hausdorff space and H a subcollection of the lattice F of all closed subsets of X, which is hereditary with respect to closed sets and stable under finite unions. The space X is called H-trivial if the co-compact topology on H coincides with the corresponding upper Kuratowski topology. We propose a description of H-trivial spaces in terms of open sieves, providing parallels and connections with Čech-complete and sieve-complete spaces. Answering a question of Nogura and Shakhmatov, we show that if X is of pointwise countable type, then X is locally compact if and only if X×Y is F-trivial (i.e., consonant) for every F-trivial space Y. A topological proof of the fact that the rationals are not Fin-trivial is given, where Fin is the lattice of finite sets. Other examples and results related to H-trivial spaces are discussed.