Realcompactness and certain types of subframes

Dube, Themba

doi:10.1007/s00012-008-2058-0

Cited by 5 publications

(1 citation statement)

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“…We note that cb-frames and weakly cb-frames have also been considered by T. Dube [12,Definition 4.5] under different names (namely tower cozshrinkable and weakly tower coz-shrinkable) as the pointfree counterparts of the cb-spaces and weak cb-spaces of Mack and Johnson [28]. In [13], weakly tower coz-shrinkable frames are called weak-cb. Our definitions above are different, closer to the classical formulations but easily seen to be equivalent to Dube's ones.…”

Section: (4) and Corollary 34(1) It Follows Thatmentioning

confidence: 99%

Normal semicontinuity and the Dedekind completion of pointfree function rings

2016

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This paper supplements an earlier one by the authors which constructed the Dedekind completion of the ring of continuous real functions on an arbitrary frame L in terms of partial continuous real functions on L. In the present paper we provide three alternative views of it, in terms of (i) normal semicontinuous real functions on L, (ii) the Booleanization of L (in the case of bounded real functions) and the Gleason cover of L (in the general case) and (iii) Hausdorff continuous partial real functions on L. The first is the normal completion and extends Dilworth's classical construction to the pointfree setting. The second shows that in the bounded case the Dedekind completion is isomorphic to the lattice of bounded continuous real functions on the Booleanization of L and that in the non-bounded case it is isomorphic to the lattice of continuous real functions on the Gleason cover of L. Finally, the third is the pointfree version of Anguelov's approach in terms of interval-valued functions. Two new classes of frames, cb-frames and weak cb-frames, emerge naturally in the first two representations. We show that they are conservative generalizations of their classical counterparts.

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Section: (4) and Corollary 34(1) It Follows Thatmentioning

confidence: 99%