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Characterising points which make \(P\)-frames. (English) Zbl 1341.06015

\(P\)-frames, that is, completely regular frames in which every cozero element is complemented, are precisely those completely regular frames the points of whose Stone-Čech compactification are all sharp points. This paper aims at a characterization of such points in terms of convergence. For that, the authors need to extend the usual notion of convergence of filters in a frame \(L\), due to B. Banaschewski and A. Pultr [Math. Proc. Camb. Philos. Soc. 108, No. 1, 63-78 (1990; Zbl 0733.54020)], to more general filters defined in the lattice \(\mathcal S(L)\) of sublocales of \(L\). This definition generalizes that of Banaschewski and Pultr in the sense that for any \(T_1\)-frame, a filter \(F\) in \(L\) converges if and only if the filter in \(\mathcal S(L)\) generated by the open sublocales induced by the elements of \(F\) converges. Clustering for these filters is also defined and then compact frames are characterised in terms of both convergence and clustering.

MSC:

06D22 Frames, locales
54A20 Convergence in general topology (sequences, filters, limits, convergence spaces, nets, etc.)
54D30 Compactness
54G10 \(P\)-spaces

Citations:

Zbl 0733.54020
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References:

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