### 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.

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