Lindelöf locales and realcompactness

Abstract

We show that a locale possesses the localic analogue of the property of realcompactness if and only if it is regular Lindelöf. Thus, the localic version of the Hewitt real-compactification, originally defined by G.Reynolds using σ-frames, is the regular Lindelöf reflection. An immediate consequence is that a space is realcompact if and only if it is the point space of a regular Lindelöf local (3·2). We point out a nice analogy between a theorem of Reynolds and Stone's classical representation theorem for boolean algebras. Finally, we show that the quasi-F cover of a compact Hausdorff space is the Stone–čech compactifications of the smallest dense Lindelöf sublocale.