Abstract:We investigate notions of $\Bbb N$-compactness for frames. We find that the analogues of equivalent conditions defining $\Bbb N$-compact spaces are no longer equivalent in the frame context. Indeed, the closed quotients of frame `$\Bbb N$-cubes' are exactly 0-dimensional Lindel\"of frames, whereas those frames which satisfy a property based on the ultrafilter condition for spatial $\Bbb N$-compactness form a much larger class, and better embody what `$\Bbb N$-compact frames' should be. This latter property is expressible without reference to maximal ideals or filters. We construct the co-reflections for both of the classes, (the `$\Bbb N$-compactifications'), which both restrict to the spatial $\Bbb N$-compactification.
Keywords: frame, locale, complete Heyting algebra, $\Bbb N$-compact
AMS Subject Classification: Primary 06A23, 06D20, 54A05, 54D20; Secondary 18B30