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Keywords:
frame; locale; complete Heyting algebra; $\Bbb N$-compact
frame; locale; complete Heyting algebra; $\Bbb N$-compact
Summary:
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 Lindelöf 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.
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