  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.
