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.


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
Full Text: DOI


[1] Baboolal, D.; Banaschewski, B., Connectedness and local connectedness of frames, J. pure appl. algebra, 70, 3-16, (1991) · Zbl 0722.54031
[2] Banaschewski, B., Local connectedness of extension spaces, Canad. J. math., 8, 395-398, (1956) · Zbl 0072.17703
[3] Banaschewski, B.; Mulvey, C.J., Stone-C̆ech compactification of locales, Houston J. math., 6, 301-312, (1980) · Zbl 0473.54026
[4] De Groot, J.; McDowell, R.H., Locally connected spaces and their compactifications, Illinois J. math., 11, 353-364, (1967) · Zbl 0147.41602
[5] Henriksen, M.; Isbell, J.R., Local connectedness in the stone-C̆ech compactification, Illinois J. math., 1, 574-582, (1957) · Zbl 0079.38604
[6] Isbell, J.R., Atomless parts of spaces, Math. scand., 31, 5-32, (1972) · Zbl 0246.54028
[7] Johnstone, P.T., Stone spaces, (1982), Cambridge University Press Cambridge · Zbl 0499.54001
[8] Joyal, A.; Tierney, M., An extension of the Galois theory of Grothendieck, () · Zbl 0541.18002
[9] Walker, R.C., The stone-C̆ech compactification, (1974), Springer Berlin · Zbl 0292.54001
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