Quasi-Menger and weakly Menger locales

Abstract

We  study the quasi-Menger and weakly Menger   properties in locales. Our definitions, which  are adapted from topological spaces by replacing subsets with sublocales, are  conservative in the sense that a topological space  is quasi-Menger (resp. weakly Menger) if and only if the locale  it determines is quasi-Menger (resp. weakly Menger). We characterize each of these types of locales in a language that does not involve sublocales. Regarding localic results that have no topological counterparts, we show that an infinitely extremally disconnected locale (in the sense of Arietta~\cite{A}) is weakly Menger if and only if its smallest dense sublocale is weakly Menger. We show that if the product of   locales is quasi-Menger (or weakly Menger) then so is each factor. Even though the localic product ${\prod_{j\in J}}\Omega(X_j)$ is not necessarily isomorphic to the locale $\Omega\big({\prod_{j\in J}}X_j\big)$, we are able to deduce as a corollary of the  localic result that  if the product of   topological spaces is weakly Menger, then so is each factor.