On the Menger and almost Menger properties in locales
DOI:
https://doi.org/10.4995/agt.2021.14915Keywords:
Menger, almost Menger, frame, locale, sublocale, spectrum of a frameAbstract
The Menger and the almost Menger properties are extended to locales. Regarding the former, the extension is conservative (meaning that a space is Menger if and only if it is Menger as a locale), and the latter is conservative for sober TD-spaces. Non-spatial Menger (and hence almost Menger) locales do exist, so that the extensions genuinely transcend the topological notions. We also consider projectively Menger locales, and show that, as in spaces, a locale is Menger precisely when it is Lindelöf and projectively Menger. Transference of these properties along localic maps (via direct image or pullback) is considered.
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