On the Order-Theoretical Foundation of a Theory of Quasicompactly Generated Spaces Without Separation Axiom

A Hausdorff space X is said to be compactly generated (a /e-space) if and only if the open subsets V of X are precisely those subsets for which K n V is open in K for all compact subsets of K of X. We interpret this property as a duality property of the lattice 0{ X) of open sets of X, This view point allows the introduction of the concept of being quasicompactly generated for an arbitrary sober space X. The methods involve the duality theory of up-complete semilattices, and certain inverse limit constructions. In the process, we verify that the new concept agrees with the classical one on Hausdorff spaces. © 1984, Australian Mathematical Society. All rights reserved