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On the local connectedness of frames. (English) Zbl 0753.54008

A result of J. de Groot and R. H. McDowell [Ill. J. Math. 11, 353-364 (1967; Zbl 0147.416)] states that if \(X\) is a Tychonov space, then any Tychonov space containing \(X\) as a dense subspace is locally connected if and only if \(X\) is locally connected and pseudocompact. The main theorem of the paper under review states that if \(M\) is a compact locally connected frame and \(L\) is a regular subframe of \(M\), then \(L\) is also locally connected. As a corollary, using also results of D. Baboolal and B. Banaschewski [J. Pure Appl. Algebra 70, No. 1/2, 3-16 (1991; Zbl 0722.54031)], the author obtains the following frame counterpart of the cited topological result: For a completely regular locale \(L\), the following are equivalent: 1) \(L\) is pseudocompact and locally connected, 2) Every completely regular locale with \(L\) as a dense sublocale is locally connected. 3) \(\beta L\) is locally connected.

MSC:

54D05 Connected and locally connected spaces (general aspects)
54F05 Linearly ordered topological spaces, generalized ordered spaces, and partially ordered spaces
06B30 Topological lattices
06B10 Lattice ideals, congruence relations
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References:

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