On a generalization of pointfree realcompactness

We study almost* realcompact frames. Spaces with this name were introduced by Schommer and Swardson. Amplifying one of their results, we show how an almost* realcompact space can be built from subspaces that have the property. If the domain of a continuous function is almost* realcompact, it does not follow that the codomain is also almost* realcompact; and vice versa. We show that if h : L → M is a dense coz-onto homomorphism whose right adjoint preserves directed joins, then L is almost* realcompactness if and only if M is almost* realcompact.