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 hγ:γL→γM such that σM⋅hγ=h⋅σL, 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 hγ is open. Furthermore, the same condition ensures that hγ 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 hυ is surjective. The instances when hβ or hλ is surjective are known. It turns out that the surjectivity of the lifted map hυ:υL→υM captures Blair's notion of υ-embedding in the sense that a subspace S of a Tychonoff space X is υ-embedded iff the lifted map υ(Oi):υ(OX)→υ(OS) is surjective, where i:S→X is the subspace embedding.