Core compactness and diagonality in spaces of open sets

Original · EN

We investigate when the space Oₓ 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 Oₓₓ with the product of the Scott topologies of Oₓ at (X,X). On the other hand, we characterize diagonality of Oₓ 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.