Core compactness and diagonality in spaces of open sets

We investigate when the space $\mathcal O_X$ of open subsets of a topological space $X$ endowed with the Scott topology is core compact. Such conditions turn out to be related to infraconsonance of $X$, which in turn is characterized in terms of coincidence of the Scott topology of $\mathcal O_X\times\mathcal O_X$ with the product of the Scott topologies of $\mathcal O_X$ at $(X,X)$. On the other hand, we characterize diagonality of $\mathcal O_X$ endowed with the Scott convergence and show that this space can be diagonal without being pretopological. New examples are provided to clarify the relationship between pretopologicity, topologicity and diagonality of this important convergence space.