Pseudocompleteness in the category of locales

Abstract

We define pseudocompleteness in the category of locales in a conservative way; so that a space is pseudocomplete in the sense of Oxtoby [24] if and only if the locale it determines is pseudocomplete. We show that a pseudocomplete locale whose Gδ-sublocales are complemented (for instance if it is scattered) is a Baire locale in the sense of Isbell [20]. Our main theorem is that products of pseudocomplete locales are pseudocomplete. Whereas every discrete space is pseudocomplete, and Boolean locales generalize discrete spaces, we demonstrate that not every Boolean locale is pseudocomplete. In [27] Pichardo-Mendoza asks whether pseudocompleteness (in topological spaces) is an invariant of closed irreducible maps. We answer this in the affirmative.