For a Hausdorff space $X$, let $ F$ be the hyperspace of all closed subsets of X and $H$ a sublattice of $ F$. Following Nogura and Shakhmatov, $X$ is said to be $H$-trivial if the upper Kuratowski topology and the co-compact topology coincide on $H$. $F$-trivial spaces are the consonant spaces first introduced and studied by Dolecki, Greco and Lechicki. In this paper, we deal with $K$-trivial spaces and Fin-trivial space, where $K$ and Fin are respectively the lattices of compact and of finite subsets of $X$. It is proved that if $C_k(X)$ is a Baire space or more generally if $X$ has ‘the moving off property' of Gruenhage and Ma, then $X$ is $K$-trivial. If $X$ is countable, then $C_p(X)$ is Baire if and only if $X$ is Fin-trivial and all compact subsets of $X$ are finite. As for consonant spaces, it turns out that every regular $K$-trivial space is a Prohorov space. This result remains true for any regular Fin-trivial space in which all compact subsets are scattered. It follows that every regular first countable space without isolated points, all compact subsets of which are countable, is Fin-nontrivial. Examples of $K$-trivial non-consonant spaces, of Fin-trivial $K$-nontrivial spaces and of countably compact Prohorov Finnontrivial spaces, are given. In particular, we show that all (generalized) Fréchet-Urysohn fans are $K$-trivial, answering a question by Nogura and Shakhmatov. Finally, we describe an example of a continuous open compact-covering mapping $f :X\to Y$, where $X$ is Prohorov and $Y$ is not Prohorov, answering a long-standing question by Topsøe.