A uniform view of localic realcompactness Original Research Article

Reynoldʹs 1977 treatment of realcompactness for a Grothendieck topos (1) requires the axiom of countable choice because it depends on cozero σ-frames, and (2) does not faithfully reflect Hewittʹs original notion since it only captures Lindelöf spaces, as was discovered some years later. Here, a choice-free version of Reynolds’ result is presented, based on suitable uniformities, together with a new variant which provides an exact extension of Hewittʹs realcompactness.