Geodesic
N-compact frames
Commentationes Mathematicae Universitatis Carolinae, Tome 32 (1991) no. 1, pp. 173-187
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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.
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.
Classification : 06A23, 06D20, 06D99, 18B30, 54A05, 54D20
Keywords: frame; locale; complete Heyting algebra; $\Bbb N$-compact 
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Schlitt, Greg M. N-compact frames. Commentationes Mathematicae Universitatis Carolinae, Tome 32 (1991) no. 1, pp. 173-187. http://geodesic.mathdoc.fr/item/CMUC_1991_32_1_a17/

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