When lifted frame homomorphisms are closed (Q439304)

The lift \(h^{\beta}: \beta L\to \beta M\) of any frame homomorphism \(h: L\to M\) to the Stone-Čech compactifications is always closed since any frame homomorphism into a compact regular frame is closed. In the present paper the authors investigate whether the lifts \(h^{\lambda}: \lambda L\to \lambda M\) and \(h^{\upsilon}: \upsilon L\to \upsilon M\) to the Lindelöf and the realcompact coreflections, respectively, are closed. Among other results, the authors show the following: {\parindent=6mm \begin{itemize}\item[(1)] For a completely regular frame \(L\), \(L\) is weakly normal iff the lift \(h^{\lambda}\) of any closed homomorphism \(h\) with domain \(L\) is closed. \item[(2)] For a completely regular frame \(L\), \(L\) is weakly normal iff the lift \(h^{\lambda}\) of any closed homomorphism \(h\) with domain \(L\) is weakly closed. \item[(3)] If the lift \(h^{\upsilon}\) of a homomorphism \(h\) with a weakly normal domain is closed, then the lift \(h^{\lambda}\) is also closed. NEWLINENEWLINE\end{itemize}} A corollary of (1) includes a new characterization of topological spaces that have Property Z [\textit{P. Zenor}, Proc. Am. Math. Soc. 23, 273--275 (1969; Zbl 0186.26702)] among Tychonoff spaces whose Hewitt realcompactification is Lindelöf.NEWLINENEWLINEGoing in the opposite direction, when does closedness descend from the lifts? The final part of the paper addresses this question and provides the following answers: {\parindent=6mm \begin{itemize}\item[(4)] If the lift \(h^{\lambda}\) of a \(\lambda\)-map \(h\) with normal codomain is closed, then \(h\) is closed. \item[(5)] A \(\lambda\)-map \(h\) between normal frames is closed iff its lift \(h^{\lambda}\) is closed. \item[(6)] If the lift \(h^{\lambda}\) of a homomorphism \(h\) with weakly normal codomain is weakly closed, then \(h\) is weakly closed. \item[(7)] If the lift \(h^{\lambda}\) of a homomorphism \(h: L\to M\) is closed and coincides with \(h^{\upsilon}\) on \(\upsilon L\), and \(\upsilon M\) is normal, then \(h^{\upsilon}\) is closed. NEWLINENEWLINE\end{itemize}}