On openness and surjectivity of lifted frame homomorphisms

Given a completely regular frame L, let, as usual, βL, λL and υL denote, respectively, the Stone–Čech compactification, the universal Lindelöfication and the Hewitt realcompactification of L. Let γ denote any of the functors β, λ or υ. It is well known that any frame homomorphism h : L → M has a unique “lift” to a frame homomorphism h γ : γ L → γ M such that σ M ⋅ h γ = h ⋅ σ L , where the σ-maps are effected by join. We find a condition on h such that if h satisfies it, then h is open iff its lift h γ is open. Furthermore, the same condition ensures that h γ is nearly open iff h is nearly open. This latter result is, in fact, a special case of a more general phenomenon. In the last part of the paper we investigate when h υ is surjective. The instances when h β or h λ is surjective are known. It turns out that the surjectivity of the lifted map h υ : υ L → υ M captures Blairʹs notion of υ-embedding in the sense that a subspace S of a Tychonoff space X is υ-embedded iff the lifted map ( O i ) υ : υ ( O X ) → υ ( O S ) is surjective, where i : S → X is the subspace embedding.