When do the upper Kuratowski topology (homeomorphically, Scott topology) and the co-compact topology coincide?

Abstract

A topology is called consonant if the corresponding upper Kuratowski topology on closed sets coincides with the co-compact topology, equivalently if each Scott open set is compactly generated. It is proved that Čechcomplete topologies are consonant and that consonance is not preserved by passage to G δ {G_\delta } -sets, quotient maps and finite products. However, in the class of the regular spaces, the product of a consonant topology and of a locally compact topology is consonant. The latter fact enables us to characterize the topologies generated by some Γ \Gamma -convergences.