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On the paracompactness of frames. (English) Zbl 0795.54033

There are some frame properties whose corresponding topological properties have important characterizations in terms of subsets that are not necessarily open. For instance, local connectedness and paracompactness are such properties. To get frame counterparts of these valuable characterizations, we turn to frame congruences, which provide sufficient tools for us to translate directly some topological properties pertaining to general subsets, especially, closed subsets of spaces. Advantages of such considerations have been shown through the study of the local connectedness of frames in [the author, J. Pure Appl. Algebra 79, No. 1, 35-43 (1992; Zbl 0753.54008)].
It is well known that the paracompactness of spaces can be characterized by employing one of the following refinements: (1) locally finite; (2) cushioned; (3) closure-preserving; (4) \(\sigma\)-locally finite open; (5) \(\sigma\)-closure preserving open; and (6) \(\sigma\)-cushioned open. Through the study of congruences, and applying the results of [C. H. Dowker and D. Papert Strauss, Symp. math. 16, Topol. insiem. gen., Gruppi topol. Gruppi di Lie, Convegni 1973/74, 93-116 (1975; Zbl 0324.54015)], we obtain the frame versions of the above classical characterizations, and thus extend the related results of [Dowker and Papert Strauss, loc. cit.], [A. Pultr and J. Ulehla, Commentat. Math. Univ. Carol. 30, No. 2, 377-384 (1989; Zbl 0672.54018)] and [S. Sun, ibid. 30, No. 1, 101-107 (1989; Zbl 0668.06009)]. The topological intuition behind our arguments concerning congruences can be easily traced by knowing the correspondence between congruences and subspaces.

MSC:

54D20 Noncompact covering properties (paracompact, Lindelöf, etc.)
06D99 Distributive lattices
54J05 Nonstandard topology
06B10 Lattice ideals, congruence relations
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