## 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

### Citations:

Zbl 0147.416; Zbl 0722.54031
Full Text:

### References:

 [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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